Constant current to constant voltage dual active bridge lcl-t resonant dc-dc converter

ABSTRACT

A power converter includes a primary H-bridge with switches and an LCL-T section with a first inductor with a first end connected to a first terminal of the primary H-bridge, a capacitor connected between a second end of the first inductor and a second terminal of the primary H-bridge, and a second inductor with a first end connected to the second end of the first inductor. The converter includes a transformer with a primary connected between a second end of the second inductor and the second terminal of the primary H-bridge, a secondary H-bridge with switches with an input connected to a secondary side of the transformer, and an output capacitor connected across output terminals of the secondary H-bridge. The primary H-bridge is fed by a DC constant current source and the output terminals of the secondary H-bridge have a regulated DC output voltage are connected to a load.

FIELD

This invention relates to resonant converters and more particularlyrelates to constant current to a constant voltage dual active bridge(“DAB”) inductor-capacitor-inductor (“LCL”) T-type resonant directcurrent (“DC”)-to-DC converter.

BACKGROUND

Resonant power converters are a popular choice for DC-DC powerconversion at high switching frequency due to their soft-switchingcapability, high efficiency, high power density, and low electromagneticinterference (“EMI”). Because of these inherent advantages, resonantconverters are widely used in various application such astelecommunications, energy storage, undersea DC distribution networks,wireless power transfer systems, battery or capacitor charging, LEDdrivers, and the like.

In long distance underwater ocean observatory systems, converters placedon the seabed are distant from the source and a constant DC currentbased power distribution is preferred because of its robustness againstcable impedance and faults. A block diagram of such a distributionnetwork is shown in FIG. 1 , where the onshore power source drives aconstant current through the trunk cable to m series connected, isolatedDC-DC converter module(s) that are contained in their respectivehermetically sealed enclosures and thus the components do not come incontact with the sea-water. IN some embodiments, utilizing theconductivity of saline water, the cable current returns to the sourcethrough seawater. Each DC-DC converter taps power from this constantcurrent source to regulate its output voltage or current.

With a current source as input, converters employ a current fed inverter(“CFI”) stage at front end that can operate with zero current switching(“ZCS”). However, achieving zero voltage switching (“ZVS”) ischallenging in CFI, limiting the switching frequency of operation. Withresonant topologies, switches in a CFI often must be rated for a valuehigher than average DC input voltage, which makes the CFI stageimpractical for low-current high-voltage systems. Hence, a voltage fedinverter stage is often used at the front end, where the DC inputvoltage varies with the load. Various topologies have been tried, butall have distinct deficiencies.

SUMMARY

A power converter includes a primary H-bridge that includessemi-conductor switches. The power converter has an LCL-T section thatincludes a first inductor L_(r) with a first end connected to a firstterminal A of the primary H-bridge, a capacitor C_(r) connected betweena second end of the first inductor L_(r) and a second terminal B of theprimary H-bridge, and a second inductor L_(g) with a first end connectedto the second end of the first inductor L_(r). The power converterincludes a transformer with a primary side connected between a secondend of the second inductor L_(g) and the second terminal B of theprimary H-bridge, a secondary H-bridge that includes semi-conductorswitches with an input connected to a secondary side of the transformer,and an output capacitor C_(f) connected across output terminals of thesecondary H-bridge. The primary H-bridge is fed by a direct current(“DC”) constant current source and the output terminals of the secondaryH-bridge are connected to a load and an output voltage of the secondaryH-bridge is regulated to maintain a constant DC output voltage.

Another embodiment of a power converter includes a primary H-bridge withfour semi-conductor switches where two of the switches are in leg A withterminal A between the switches in leg A and two of the switches are inleg B with terminal B between the switches in leg B. Terminal A andterminal B form an output of the primary H-bridge. The power converterincludes an LCL-T section that includes a first inductor L_(r) with afirst end connected to terminal A, a capacitor C_(r) connected between asecond end of the first inductor L_(r) and terminal B, and a secondinductor L_(g) with a first end connected to the second end of the firstinductor L_(r). The power converter includes a transformer with aprimary side connected between a second end of the second inductor L_(g)and terminal B where the transformer has a turns ratio n. The powerconverter includes a secondary H-bridge that includes semi-conductorswitches with an input connected to a secondary side of the transformerwhere two of the switches are in leg D with terminal D between the twoswitches of leg D and two of the switches are in leg E with terminal Ebetween the two switches of leg E. Terminal D and terminal E form anoutput of the secondary H-bridge. The power converter includes an outputcapacitor C_(f) connected across terminal D and terminal E. The primaryH-bridge is fed by a DC constant current source and terminals D and Eare connected to a load and an output voltage across terminals D and Eis regulated to maintain a constant DC output voltage.

Another bidirectional power converter includes a primary H-bridge thatincludes four semi-conductor switches where two of the switches are inleg A with terminal A between the switches in leg A and two of theswitches are in leg B with terminal B between the switches in leg B.Terminal A and terminal B form an output of the primary H-bridge. Thebidirectional power converter includes an LCL-T section that includes afirst inductor L_(r) with a first end connected to terminal A, acapacitor C_(r) connected between a second end of the first inductorL_(r) and terminal B, and a second inductor L_(g) with a first endconnected to the second end of the first inductor L_(r). Thebidirectional power converter includes a transformer with a primary sideconnected between a second end of the second inductor L_(g) and terminalB. The transformer has a turns ratio n. The bidirectional powerconverter includes a secondary H-bridge that includes semi-conductorswitches with an input connected to a secondary side of the transformerwhere two of the switches are in leg D with terminal D between the twoswitches of leg D and two of the switches are in leg E with terminal Ebetween the two switches of leg E. Terminal D and terminal E form anoutput of the secondary H-bridge.

The bidirectional power converter includes an output capacitor C_(ƒ)connected across terminal D and terminal E. The primary H-bridge is fedby a DC constant current source and terminals D and E are connected to aload and an output voltage across terminals D and E is regulated tomaintain a constant DC output voltage. The switches of the primaryH-bridge are arranged in a leg A and a leg B and the switches of thesecondary H-bridge are arranged in a leg D and a leg E where theswitches of the primary H-bridge are operated with symmetrical phaseshift modulation with leg A leading leg B by an angle φ_(AB), theswitches of the secondary H-bridge are operated with symmetrical phaseshift modulation with leg D leading leg E by an angle φDE, an anglebetween leg A and leg D is angle φAD, and a power flow direction fromthe primary H-bridge to the secondary H-bridge is dependent on a phaseangle φ_(PS), which is:

$\varphi_{PS} = \varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}.$

BRIEF DESCRIPTION OF THE DRAWINGS

In order that the advantages of the invention will be readilyunderstood, a more particular description of the invention brieflydescribed above will be rendered by reference to specific embodimentsthat are illustrated in the appended drawings. Understanding that thesedrawings depict only typical embodiments of the invention and are nottherefore to be considered to be limiting of its scope, the inventionwill be described and explained with additional specificity and detailthrough the use of the accompanying drawings, in which:

FIG. 1 is schematic block diagram illustrating one embodiment of anunderwater DC distribution network fed from an onshore DC currentsource;

FIG. 2 is a schematic block diagram illustrating one embodiment of (a)an LCL-T resonant DC-DC converter topology and (b) primary sidemodulation;

FIG. 3 is (a) a schematic block diagram illustrating one embodiment ofan equivalent circuit of the LCL-T resonant converter of FIG. 2(a) and asimplified equivalent circuit for steady state analysis usingfundamental harmonic approximation;

FIG. 4 is a diagram illustrating a steady state normalized DC outputvoltage (Vout_norm) versus a normalized switching frequency (F), fordifferent load (Q), (a) with g = 1 and (b) with g = 0.3;

FIG. 5 is a schematic block diagram illustrating one embodiment of (a)an equivalent circuit of the tank, at an operating point of F = 1 and g= 1 and (b) a phasor diagram for the AC signals from the equivalentcircuit of (a);

FIG. 6 is a diagram illustrating quality factor of the tank elements(top) and normalized VA rating of the tank (bottom) with respect totransformer turns ratio;

FIG. 7 is a diagram of steady state DC output voltage (Vout) of thesecondary H-bridge versus load power (Pout) for a result from analysisand a simulation result with diode-bridge;

FIG. 8 is (a) a schematic block diagram illustrating one embodiment of adual active bridge (“DAB”) LCL-T resonant DC-DC converter topology, and(b) corresponding modulation waveforms;

FIG. 9 is a diagram illustrating steady state DC output voltage (Vout)of the secondary H-bridge versus load power (Pout) for analysis results,simulation results with a diode-bridge, and simulation results with aDAB;

FIG. 10 includes oscilloscope waveforms from a hardware test setup withsteady state operating waveforms with a diode-bridge on the secondaryH-bridge with φAB = 120°, (a) at 50 W and (b) at 500 W where CH1:ν_(AB), CH2: i_(t), CH3: ν_(DE), CH4: i_(D);

FIG. 11 includes oscilloscope waveforms from the hardware test setupwith steady state operating waveforms with active bridge on thesecondary side with φ_(AB) = 120°, φ_(DE) = 180° and φ_(AD) = 60°, (a)at 50 W and (b) at 500 W where CH1: ν_(AB), CH2: it, CH3: ν_(DE), CH4:iD;

FIG. 12 includes oscilloscope waveforms from the hardware test setupwith gate-source and drain-source waveforms of bottom switch(es) of allthe legs of the primary H-bridge at (a) 50 W and (b) 500 W, andsecondary H-bridge at (c) 50 W and (d) 500 W, operated with φ_(AB) =120°, φ_(DE) = 180° and φ_(AD) = 60°, showing ZVS operation;

FIG. 13 is a diagram illustrating steady state DC output voltage (Vout)of the secondary H-bridge versus load power (Pout) with φ_(AB) = 120°with analysis results, experimental results with a diode-bridge on thesecondary H-bridge, and experimental results with an active bridge onthe secondary H-bridge;

FIG. 14 is a diagram illustrating variation in control angle φ_(AB) toregulate Vout at a fixed value of 150 V, versus load power with analysisresults, experimental results with a diode-bridge on the secondaryH-bridge, and experimental results with an active bridge on thesecondary H-bridge;

FIG. 15 DAB LCL-T operation under load transient from 350 W to 400 W andback to 350 W with fixed φAB = 115° with CH1: output current (I_(load)),CH2: input voltage (Vin) and CH4: output voltage (Vout);

FIG. 16 is a diagram illustrating a comparison of analytical andexperimentally measured rms values of tank signals of the DAB LCL-Tconverter over a load range wherein the top plot pane: It, rms and IR,rms, and bottom plot pane: VCr,rms, and where solid lines are foranalytical and dots are for experimental results;

FIG. 17 is a diagram illustrating power loss distribution among thecomponents of DAB LCL-T converter at full load (500 W);

FIG. 18 is a diagram illustrating efficiency of the LCL-T converter overa load range, operating at fixed control angle φ_(AB) = 120° withexperimental result with a diode-bridge on the secondary H-bridge andexperimental result with an active bridge on the secondary of theH-bridge;

FIG. 19 is a schematic block diagram illustrating one embodiment of (a)a system level block diagram of undersea DC constant currentdistribution network with power branching units (“PBUs”) and (b)components within a PBU catering to critical loads;

FIG. 20 is a schematic block diagram illustrating one embodiment ofpower flow through a PBU (a) from the trunk cable to the load undernormal operating conditions, and (b) from an auxiliary source to theload under a cable fault condition;

FIG. 21 is (a) a schematic block diagram illustrating one embodiment ofa DAB LCL-T resonant DC-DC converter topology and (b) associatedmodulation waveforms;

FIG. 22 is a schematic block diagram illustrating one embodiment of (a)an equivalent circuit of the DAB LCL-T resonant DC-DC converter and (b)a fundamental AC equivalent circuit of the loaded resonant tank;

FIG. 23 is a phasor diagram for the AC signals from the equivalentcircuit of FIG. 22(b);

FIG. 24 is a diagram illustrating variations of φ_(SL), φ_(e) andnormalized power over the range of φ_(PS), for forward and reverse powerflow;

FIG. 25 is a diagram illustrating normalized VA rating of the tank forvarious transformer turns ratios (n);

FIG. 26 includes oscilloscope waveforms from a DAB LCL-T converterhardware test setup illustrating steady state operating waveforms (a) at50 W and (b) 500 W for forward power transfer from a 1 A current source(I_(g)) with φ_(AB) = 120°, φ_(DE) = 180°, and φ_(AD) = 60° and (c) at50 W and (d) at 500 W for reverse power transfer from a 150 V voltagesource (V_(g)) with φ_(AB) = 120°, φ_(DE) = 180°, and φ_(AD) = -120°where CH1: V_(AB), CH2: i_(t), CH3: V′ _(DE), and CH4: i’_(R);

FIG. 27 are diagrams illustrating steady state DC output versus loadpower for (a) output voltage in forward power and (b) output current inreverse power with analysis results and experimental results;

FIG. 28 is a diagram illustrating variation in control angle φ_(AB) overthe range of load power with analytical results, regulating outputvoltage (vout) at 150 V for forward power flow, and regulating outputcurrent (I_(g)) for reverse power transfer;

FIG. 29 are diagrams illustrating a comparison of analytical andexperimentally measured rms values of tank signals plotted against loadpower for (a) forward power and (b) reverse power where solid linesrepresent analytical expressions and dots represent experimental resultsoperating with φ_(AB) = 120°;

FIG. 30 is a diagram illustrating experimentally measured efficiency ofthe converter versus load power for forward and reverse power, whileregulating its outputs at 150 V and 1 A, respectively;

FIG. 31 is a schematic block diagram illustrating one embodiment of aDAB LCL-T resonant DC-DC converter topology with ZVS branches;

FIG. B1 is diagram illustrating the primary H-bridge inverter’s outputvoltage (ν_(AB)) and current waveform (left) and DC side input currentwaveform (right);

FIG. C1 is a diagram illustrating percentage variation in DC outputvoltage (V_(out)) for variation in tank elements for variation in Lr,variation in Cr, and for variation in Lg and the solid lines are foranalytical results and dots are for simulation results;

FIG. D1 is (a) a schematic block diagram illustrating one embodiment ofthe secondary H-bridge with a capacitive filter on the DC side and (b)its AC and DC side voltage and current waveforms; and

FIG. E1 is a diagram illustrating percentage variation in DC output withrespect to variation in resonant tank elements where solid lines are forforward power transfer and dotted lines are for reverse power flowoperation showing plots for L_(r), C_(r) and Lg.

DETAILED DESCRIPTION

Reference throughout this specification to “one embodiment,” “anembodiment,” or similar language means that a particular feature,structure, or characteristic described in connection with the embodimentis included in at least one embodiment. Thus, appearances of the phrases“in one embodiment,” “in an embodiment,” and similar language throughoutthis specification may, but do not necessarily, all refer to the sameembodiment, but mean “one or more but not all embodiments” unlessexpressly specified otherwise. The terms “including,” “comprising,”“having,” and variations thereof mean “including but not limited to”unless expressly specified otherwise. An enumerated listing of itemsdoes not imply that any or all of the items are mutually exclusiveand/or mutually inclusive, unless expressly specified otherwise. Theterms “a,” “an,” and “the” also refer to “one or more” unless expresslyspecified otherwise.

Furthermore, the described features, structures, or characteristics ofthe invention may be combined in any suitable manner in one or moreembodiments. In the following description, numerous specific details areprovided, such as examples of programming, software modules, userselections, network transactions, database queries, database structures,hardware modules, hardware circuits, hardware chips, etc., to provide athorough understanding of embodiments of the invention. One skilled inthe relevant art will recognize, however, that the invention may bepracticed without one or more of the specific details, or with othermethods, components, materials, and so forth. In other instances,well-known structures, materials, or operations are not shown ordescribed in detail to avoid obscuring aspects of the invention.

As used herein, a list with a conjunction of “and/or” includes anysingle item in the list or a combination of items in the list. Forexample, a list of A, B and/or C includes only A, only B, only C, acombination of A and B, a combination of B and C, a combination of A andC or a combination of A, B and C. As used herein, a list using theterminology “one or more of” includes any single item in the list or acombination of items in the list. For example, one or more of A, B and Cincludes only A, only B, only C, a combination of A and B, a combinationof B and C, a combination of A and C or a combination of A, B and C. Asused herein, a list using the terminology “one of” includes one and onlyone of any single item in the list. For example, “one of A, B and C”includes only A, only B or only C and excludes combinations of A, B andC.

A power converter includes a primary H-bridge that includessemi-conductor switches. The power converter has an LCL-T section thatincludes a first inductor L_(r) with a first end connected to a firstterminal A of the primary H-bridge, a capacitor C_(r) connected betweena second end of the first inductor L_(r) and a second terminal B of theprimary H-bridge, and a second inductor L_(g) with a first end connectedto the second end of the first inductor L_(r). The power converterincludes a transformer with a primary side connected between a secondend of the second inductor L_(g) and the second terminal B of theprimary H-bridge, a secondary H-bridge that includes semi-conductorswitches with an input connected to a secondary side of the transformer,and an output capacitor C_(f) connected across output terminals of thesecondary H-bridge. The primary H-bridge is fed by a direct current(“DC”) constant current source and the output terminals of the secondaryH-bridge are connected to a load and an output voltage of the secondaryH-bridge is regulated to maintain a constant DC output voltage.

In some embodiments, a switching frequency of the switches of theprimary H-bridge and the secondary H-bridge is selected to be within 15percent of a resonant frequency of the LCL-T section. In otherembodiments, a ratio g of the first inductor L_(r) and the secondinductor L_(g) is set to be within a range of 0.2 to 5 (g = 0.2 to 5).In other embodiments, the switches of the primary H-bridge are arrangedin a leg A and a leg B and the switches of the secondary H-bridge arearranged in a leg D and a leg E and the switches of the primary H-bridgeare operated with symmetrical phase shift modulation with leg A leadingleg B by an angle φ_(AB), the switches of the secondary H-bridge areoperated with symmetrical phase shift modulation with leg D leading legE by an angle φ_(DE), an angle between leg A and leg D is angle φ_(AD),and the output voltage of the secondary H-bridge is maintained at aconstant voltage by controlling angle φ_(AB), angle φ_(DE), and angleφ_(AD).

In other embodiments, a relationship between angle φ_(AB), angle φ_(DE),and angle φ_(AD) is:

$\varphi_{AD} = \frac{\varphi_{AB}}{2} + \frac{\pi}{2} - \frac{\varphi_{DE}}{2}.$

In other embodiments, angle φ_(DE) is 180 degrees and a relationshipbetween angle φ_(AB) and angle φ_(AD) is:

$\varphi_{AD} = \frac{\varphi_{AB}}{2}.$

In other embodiments, angle φ_(AB) is controlled as a function of theoutput voltage of the secondary H-bridge and angle φ_(AD) is controlledto be half the angle φ_(AB), or angle φ_(AD) is controlled as a functionof the output voltage of the secondary H-bridge and angle φ_(AB) iscontrolled to be twice the angle φ_(AD).

In some embodiments, power flow is bidirectional. In other embodiments,the switches of the primary H-bridge are arranged in a leg A and a legB, the switches of the secondary H-bridge are arranged in a leg D and aleg E and the switches of the primary H-bridge are operated withsymmetrical phase shift modulation with leg A leading leg B by an angleφ_(AB), the switches of the secondary H-bridge are operated withsymmetrical phase shift modulation with leg D leading leg E by an angleφ_(DE), an angle between leg A and leg D is angle φ_(AD), and a powerflow direction from the primary H-bridge to the secondary H-bridge isdependent on a phase angle φ_(PS), which is:

$\varphi_{PS} = \varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}.$

In other embodiments, φ_(PS) is within the range [0, _(Π)] for forwardpower flow where input current ^(I)1 to the primary H-bridge and outputcurrent I₂ from the secondary H-bridge are positive, and φ_(ps) iswithin the range [-Π, 0] for reverse power flow where I₁ and I₂ are bothnegative. In other embodiments, angle φ_(DE) is 180 degrees and arelationship between angle φ_(AB) and angle φ_(AD) is

$\varphi_{AD} = \frac{\varphi_{AB}}{2}$

for forward power flow, and

$\varphi_{AD} = \frac{\varphi_{AB}}{2} -$

180° for reverse power flow.

In other embodiments, for forward power flow, angle φ_(AB) is either setto a fixed value or controlled as a function of the output voltage ofthe secondary H-bridge and angle φ_(AD) is controlled to be half theangle φ_(AB), for reverse power flow, angle φ_(AB) is either set to afixed value or controlled as a function of the input current to theprimary H-bridge and angle φ_(AD) is controlled to be

$\varphi_{AD} = \frac{\varphi_{AB}}{2} - 180^{\circ}.$

For forward power flow, angle φ_(AD) is either set to a fixed value orcontrolled as a function of the output voltage of the secondary H-bridgeand angle φ_(AB) is either set to a fixed value or controlled to betwice the angle φ_(AD), or for reverse power flow, angle φ_(AD) iscontrolled as a function of the input current to the primary H-bridgeand angle φ_(AB) is controlled to be φ_(AB) = 2(φ_(AD) + 180°).

In some embodiments, a turns ratio n of the transformer is set at anoptimal turns ratio n_(opt:)

$n_{opt} = \frac{P_{load\_ max}sin( \frac{\varphi_{AB}}{2} )}{V_{2}I_{g}},$

where the switches of the primary H-bridge are operated with symmetricalphase shift modulation with leg A leading leg B by an angle φ_(AB),P_(load_) _(max) is a maximum load condition, I_(g) is a DC constantsource current, and V₂ is a constant output voltage of the secondaryH-bridge. In other embodiments, the power converter includes an inputcapacitor C_(in) connected across input terminals of the primaryH-bridge.

Another embodiment of a power converter includes a primary H-bridge withfour semi-conductor switches where two of the switches are in leg A withterminal A between the switches in leg A and two of the switches are inleg B with terminal B between the switches in leg B. Terminal A andterminal B form an output of the primary H-bridge. The power converterincludes an LCL-T section that includes a first inductor L_(r) with afirst end connected to terminal A, a capacitor C_(r) connected between asecond end of the first inductor L_(r) and terminal B, and a secondinductor L_(g) with a first end connected to the second end of the firstinductor L_(r). The power converter includes a transformer with aprimary side connected between a second end of the second inductor L_(g)and terminal B where the transformer has a turns ratio n. The powerconverter includes a secondary H-bridge that includes semi-conductorswitches with an input connected to a secondary side of the transformerwhere two of the switches are in leg D with terminal D between the twoswitches of leg D and two of the switches are in leg E with terminal Ebetween the two switches of leg E. Terminal D and terminal E form anoutput of the secondary H-bridge. The power converter includes an outputcapacitor C_(f) connected across terminal D and terminal E. The primaryH-bridge is fed by a DC constant current source and terminals D and Eare connected to a load and an output voltage across terminals D and Eis regulated to maintain a constant DC output voltage.

In some embodiments, a switching frequency of the switches of theprimary H-bridge and the secondary H-bridge is selected to be within 15percent of a resonant frequency of the LCL-T section and a ratio g ofthe first inductor L_(r) and the second inductor L_(g) is within a rangeof 0.2 to 5 (g = 0.2 to 5). In other embodiments, the switches of theprimary H-bridge are arranged in a leg A and a leg B and the switches ofthe secondary H-bridge are arranged in a leg D and a leg E. The switchesof the primary H-bridge are operated with symmetrical phase shiftmodulation with leg A leading leg B by an angle φ_(AB), the switches ofthe secondary H-bridge are operated with symmetrical phase shiftmodulation with leg D leading leg E by an angle φ_(DE), an angle betweenleg A and leg D is angle φ_(AD), and the output voltage of the secondaryH-bridge is maintained at a constant voltage by controlling angleφ_(AB), angle φ_(DE), and angle φ_(AD) where a relationship betweenangle φ_(AB), angle φ_(DE), and angle φ_(AD) is:

$\varphi_{AD} = \frac{\varphi_{AB}}{2} + \frac{\pi}{2} - \frac{\varphi_{DE}}{2}.$

In other embodiments, angle φ_(DE) is 180 degrees and a relationshipbetween angle φ_(AB) and angle φ_(AD) is

$\varphi_{AD} = \frac{\varphi_{AB}}{2},$

and angle φ_(AB) is controlled as a function of the output voltage ofthe secondary H-bridge and angle φ_(AD) is controlled to be half theangle φ_(AB) or angle φ_(AD)is controlled as a function of the outputvoltage of the secondary H-bridge and angle φ_(AB) is controlled to betwice the angle φ_(AD).

In some embodiments, power flow is bidirectional and the switches of theprimary H-bridge are arranged in a leg A and a leg B, the switches ofthe secondary H-bridge are arranged in a leg D and a leg E and theswitches of the primary H-bridge are operated with symmetrical phaseshift modulation with leg A leading leg B by an angle φ_(AB), theswitches of the secondary H-bridge are operated with symmetrical phaseshift modulation with leg D leading leg E by an angle φ_(DE), an anglebetween leg A and leg D is angle φ_(AD), and a power flow direction fromthe primary H-bridge to the secondary H-bridge is dependent on a phaseangle φ_(PS), which is:

$\varphi_{PS} = \varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}.$

Another bidirectional power converter includes a primary H-bridge thatincludes four semi-conductor switches where two of the switches are inleg A with terminal A between the switches in leg A and two of theswitches are in leg B with terminal B between the switches in leg B.Terminal A and terminal B form an output of the primary H-bridge. Thebidirectional power converter includes an LCL-T section that includes afirst inductor L_(r) with a first end connected to terminal A, acapacitor C_(r) connected between a second end of the first inductorL_(r) and terminal B, and a second inductor L_(g) with a first endconnected to the second end of the first inductor L_(r). Thebidirectional power converter includes a transformer with a primary sideconnected between a second end of the second inductor L_(g) and terminalB. The transformer has a turns ratio n. The bidirectional powerconverter includes a secondary H-bridge that includes semi-conductorswitches with an input connected to a secondary side of the transformerwhere two of the switches are in leg D with terminal D between the twoswitches of leg D and two of the switches are in leg E with terminal Ebetween the two switches of leg E. Terminal D and terminal E form anoutput of the secondary H-bridge.

The bidirectional power converter includes an output capacitor C_(ƒ)connected across terminal D and terminal E. The primary H-bridge is fedby a DC constant current source and terminals D and E are connected to aload and an output voltage across terminals D and E is regulated tomaintain a constant DC output voltage. The switches of the primaryH-bridge are arranged in a leg A and a leg B and the switches of thesecondary H-bridge are arranged in a leg D and a leg E where theswitches of the primary H-bridge are operated with symmetrical phaseshift modulation with leg A leading leg B by an angle φ_(AB), theswitches of the secondary H-bridge are operated with symmetrical phaseshift modulation with leg D leading leg E by an angle φ_(DE), an anglebetween leg A and leg D is angle φ_(AD), and a power flow direction fromthe primary H-bridge to the secondary H-bridge is dependent on a phaseangle φ_(PS), which is:

$\varphi_{PS} = \varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}.$

As presented herein, an LCL-T resonant network based DC-DC converter isanalyzed and is shown that with suitable design, this converter canproduce a load independent, constant output voltage characteristic whenpowered from a constant DC current source input.

I. Steady State Modeling and Analysis

FIG. 2 is a schematic block diagram illustrating one embodiment of anLCL-T resonant DC-DC converter topology and primary side modulation.FIG. 2(a) shows an LCL-T resonant tank based topology that convertsconstant DC input current to constant DC output voltage. MOSFETs Q₁ - Q₄form the primary side inverter that operates with symmetrical phaseshift modulation with leg A leading leg B by an angle φ_(AB), as shownin FIG. 2(b). This inverter translates the DC bus voltage V_(in) to aquasi-square wave _(VAB) and drives the LCL-T resonant tank networkformed by inductors L_(r), L_(g) and capacitor C_(r). The resonant tankis followed by an n:1 isolation transformer, and the output of which isrectified by a secondary side diode bridge rectifier that includes ofdiodes D₁ - D₄. The final DC output is filtered through filter capacitorC_(f) before going to the load, which is represented as a resistorR_(load.) For the analysis to follow, it is assumed that all thecomponents are ideal and lossless.

With fundamental harmonics approximation (“FHA”), the converter shown inFIG. 2(a) can be drawn as the equivalent circuit shown in FIG. 3(a),where:

$I_{g} = \langle i_{in} \rangle = \frac{2I_{t}}{\pi}\sin( \frac{\varphi_{AB}}{2} )\cos( \varphi_{in} )$

$v_{AB,1} = \frac{4}{\pi}V_{in}\sin( \frac{\varphi_{AB}}{2} )\cos( {\omega_{s}t} )$

$I_{load} = \langle i_{o} \rangle = \frac{2}{\pi}{I^{\prime}}_{R}$

and the AC equivalent load resistance is given by:

${R^{\prime}}_{e} = \frac{8}{\pi^{2}}R_{load,}R_{e} = \frac{n^{2}8}{\pi^{2}}R_{load}.$

In equations (1) - (4), ω_(s) is the angular switching frequency, andfor nomenclature definition, the average value of signal x isrepresented by <X>, the amplitude of the AC side signal xy isrepresented by X_(y) and the signal or parameter x reflected to thesecondary side of the transformer are expressed with a prime (x′). Inequation (1), φ_(in) is the angle between fundamental component ofprimary side inverter output voltage and current which is given as:

φ_(in) = ∠Z_(in),

where Z_(in) is input impedance of the loaded resonant tank, seen fromthe primary inverter side, as depicted in FIG. 3(b). FIG. 3(b) shows thesimplified AC equivalent circuit of the converter, reflected to theprimary side of the transformer.

From the circuit in FIG. 3(b), the output to input voltage transferfunction can be derived as:

$\frac{v_{o,1}(s)}{v_{AB,1}(s)} = \frac{1}{1 + ( {1 + g} )\frac{s}{Q\omega_{o}} + \frac{s^{2}}{\omega_{o}^{2}} + g\frac{s^{3}}{Q\omega_{o}^{3}}},$

with the parameters ae defined as:

$\omega_{o} = \frac{1}{\sqrt{L_{r}C_{r}}},Z_{o} = \sqrt{\frac{L_{r}}{C_{r}}},g = \frac{L_{g}}{L_{r}},Q = \frac{R_{e}}{Z_{o}},F = \frac{f_{s}}{f_{o}},$

where, Z_(o) is the characteristic impedance of the resonant tank, Q isthe quality factor of the loaded tank, ƒ_(s) is the switching frequencyof operation, ƒ_(o) is the resonant frequency of L_(r) and C_(r), andω_(o) is the angular resonant frequency.

The amplitude of the AC voltages in FIG. 3(b) are given in terms of DCinput and output voltage as:

$| v_{AB,1} | = \frac{4}{\pi}V_{in}sin\mspace{6mu}( \frac{\varphi_{AB}}{2} ),| v_{o,1} | = \frac{4n}{\pi}V_{out}.$

For systems with constant DC voltage source, the DC output voltage canbe found using equation (6) and equation (8), evaluating the magnitudefrom equation (6) with s = jω_(s) and is given as:

$ V_{out} \middle| {}_{DC\_ V_{in}\mspace{6mu}} = \mspace{6mu}\frac{V_{in}}{n}\mspace{6mu}\frac{Q\mspace{6mu}\sin\mspace{6mu}( \frac{\varphi AB}{2} )}{\sqrt{Q^{2}( {1 - F^{2}} )^{2} + \lbrack {( {1 + g} )F - gF^{3}} \rbrack^{2}}}. $

However, for systems with DC current source, V_(in) is dependent on loadand expression of Vout from equation (9) cannot be used as it is. Theoutput voltage for such system is derived from the equivalent circuitsshown in FIG. 3 .

The AC active power drawn from the inverter, which equals the DC inputand output power of the lossless converter, are given as:

$P_{AC}\mspace{6mu} = \,\frac{V_{AB,1,rms}^{2}}{| z_{in} |}\cos( \varphi_{in} )\,,\mspace{6mu} P_{in}\mspace{6mu} = \, V_{in}I_{g},\mspace{6mu} P_{out}\mspace{6mu} = \mspace{6mu}\frac{V_{out}^{2}}{R_{load}},$

where, V_(AB),₁,_(rms) is the rms value of the fundamental component ofinverter output voltage v_(AB,1), as given in equation (2). Withlossless power conversion, from equation (10), the input voltage can beexpressed as:

$V_{in}\mspace{6mu} = \mspace{6mu}\frac{V_{out}^{2}}{I_{g}R_{load}}.$

Equating the DC output power to the AC active power in equation (10) andusing expressions from equations (1), (2) and (11), the DC outputvoltage can be derived as:

$V_{out}\mspace{6mu} = \mspace{6mu}\frac{\pi^{2}}{8n}\frac{Z_{o}I_{g}}{\sin( \frac{\varphi AB}{2} )}\sqrt{\frac{Q}{\cos( {\varphi in} )}\frac{| Z_{in} |}{Z_{o}}},$

and the input impedance, whose derivation is provided in the Appendix,is given by:

$Z_{in}\mspace{6mu} = \mspace{6mu}\frac{Z_{o}}{\lbrack {( {1 - gF^{2}} )^{2} + F^{2}Q^{2}} \rbrack}\lbrack {Z_{R}\mspace{6mu} + \, jZ_{I}} \rbrack,$

where, expression for Z_(R) and Z_(I) are presented in the Appendix.

The analysis presented in this section establishes the steady staterelations between DC input and output for an LCL-T resonant converter,with its dependence on tank parameters and operating point, which isused in next section for design of the converter.

II. Design of LCL-T Resonant Converter

The DC output voltage of the converter, derived in equation (12), isdependent on the resonant tank parameters, load, operating frequencyetc. In this section, it will be shown how the converter is designed,with proper choice of operating point, to achieve load independentoutput voltage from constant current input. With further analysis, adesign method to optimize tank components and transformer turns ratio,is also presented in this section.

A. Operating Point Selection

From the expression of DC output voltage in equation (12), it can benormalized to be expressed as:

$V_{out\_ norm}\mspace{6mu} = \mspace{6mu}\sqrt{\frac{Q}{\cos( {\varphi in} )}\mspace{6mu}| Z_{in\_ norm} |},$

which is normalized with a base voltage defined as:

$V_{base}\mspace{6mu} = \mspace{6mu}\frac{\pi^{2}}{8n}\frac{Z_{o}I_{g}}{\sin( \frac{\varphi AB}{2} )},$

and the normalized Z_(in) is defined as:

$Z_{in\_ norm}\mspace{6mu} = \,\frac{Z_{in}}{Z_{o}}.$

The normalized DC output voltage (V_(out_norm)) from equation (14) isplotted against normalized switching frequency (F) in FIG. 4(a) forvarious loads (Q), with g = 1. It can be seen that V_(out)__(norm)becomes independent of Q if the switching frequency of operation isselected to be equal to the resonant frequency, i.e. with F = 1. Underthis operating condition, the expression of output voltage from equation(12) is now given as:

$ V_{out} \middle| {}_{F = 1}\mspace{6mu} = \,\frac{\pi^{2}}{8n}\frac{Z_{o}I_{g}}{\sin( \frac{\varphi AB}{2} )}. $

From equation (17), it can be seen that with F = 1, Vout is alsoindependent of g (L_(g)), which is shown by V_(out)__(norm) versus Fplot in FIG. 4(b), for various Q, with an arbitrarily chosen value of g= 0.3. A special case of g = 0 makes it a parallel resonant converter,achieving load independent output voltage from a constant current sourceinput. Where the switching frequency is at or near the resonantfrequency of the tank, the output voltage is independent of g for afairly wide range. In some embodiments, g is chosen to be within a rangeof 0.2 to 5. In other embodiments, g is chosen to be within a tighterrange of 0.8 to 1.2. In other embodiments, g is chosen to be one (g =1).

From the plots in FIG. 4 , it can be observed that the output voltage isalmost load independent within ±10% of F = 1. So, it is possible tooperate the converter with a small variation in F around 1, in additionto phase shift control. In some examples, the switching frequency ischosen to be within a range of 15% of the resonant frequency of the LCLtank. In other embodiments, the switching frequency is chosen to bewithin a range of 10 percent of the resonant frequency. In otherembodiments, the switching frequency is chosen to be at the resonantfrequency. However, part variations may result in the switchingfrequency being a little different than the resonant frequency. However,with a small variation limit in F, the transient response can getlimited, depending on the magnitude of load transient, due to low marginfrom steady state operating point to controller output limit. Moreover,since the converters are part of a system of converters with commonsource, if they are controlled though F variation then differentconverters will operate at different switching frequencies, depending ontheir individual loads, which will introduce low frequency (differencein frequency among converters) ripple component injected to the sourcewhich is challenging for filter design. Hence all the converters aredesigned to operate at fixed frequency and controlled through phaseshift modulation.

With F = 1, the tank input impedance (Zin) from equation (13) can bederived as:

$ Z_{in} \middle| {}_{F = 1}\mspace{6mu} = \mspace{6mu}\frac{Z_{o}}{\sqrt{Q^{2} + ( {1 - g} )^{2}}}\angle\tan^{- 1}( \frac{1 - g}{Q} ). $

From this expression of Z_(in), if g < 1, φ_(in) is positive and thusZ_(in) becomes inductive, which can help achieving ZVS for the primaryside inverter switches. However, a non-zero φ_(in) puts a restriction onminimum power operation of the converter for which its output can beregulated. Hence, g is selected to be equal to unity and with g = 1,Z_(in) becomes resistive, making the primary side inverter operate atunity power factor (“UPF”), considering FHA. With F = 1 and g = 1,Z_(in) from equation (18) can be given as:

$ Z_{in} \middle| {}_{F = 1,\, g = 1} = \frac{Z_{O}^{2}}{R_{2}}\angle 0^{o}. $

B. Derivation of Tank Signals

With the selected operating point of F = 1 and g = 1, the tank ACcircuit in FIG. 3(b) can be simplified and redrawn as shown in FIG.5(a). This circuit is solved analytically to derive the AC signal of thetank. The solutions are provided in this section. The phasor diagram ofthe AC quantities from FIG. 5(a) is drawn in FIG. 5(b), considering thefundamental component of _(VAB) as reference.

From the equivalent circuit in FIG. 3(a), the current (it) in theinductor L_(r) can be found by balancing the DC input current andrectified average current at the primary side inverter. Since theinverter is operating at UPF, it can be expressed as:

$i_{t}(t)\mspace{6mu} = \mspace{6mu}\frac{\pi}{2}\frac{I_{g}}{\sin( \frac{\varphi_{AB}}{2} )}\cos\,( {\omega_{s}t} ).$

The resonant tank AC output voltage Vo_(,1) can be found from thefundamental component of AC input voltage to the secondary side diodebridge rectifier, reflected to the transformer primary side, and isgiven by:

$v_{o,1}(t)\mspace{6mu} = \,\frac{4n}{\pi}V_{out}\cos( {\omega_{s}t - \frac{\pi}{2}} ).$

Similarly, the current (i_(R)) in the load side inductor L_(g),connected to the secondary side diode bridge rectifier, can be given by:

$i_{R}(t)\mspace{6mu} = \mspace{6mu}\frac{\pi}{2n}I_{load}\cos( {\omega_{s}t - \frac{\pi}{2}} ).$

The voltage across and current through the resonant capacitor are givenby:

$v_{Cr}(t)\, = \,\frac{4n}{\pi}V_{out}\sqrt{1\mspace{6mu} + \mspace{6mu}\frac{1}{Q^{2}}}\mspace{6mu}\cos( {\omega_{s}t\, - \,\tan^{- 1}\mspace{6mu} Q} ),$

$i_{Cr}(t)\, = \,\frac{4n}{\pi}\frac{V_{out}}{Z_{o}}\sqrt{1\mspace{6mu} + \mspace{6mu}\frac{1}{Q^{2}}}\mspace{6mu}\cos( {\omega_{s}t\, - \,\tan^{- 1}\mspace{6mu}( \frac{1}{Q} )} ).$

The detailed derivations of the tank signals can be found in theAppendix.

C. RMS Values and VA of Tank Components

From equations (20) - (24) the rms values of tank signals can be foundout as:

$I_{t,rms}\mspace{6mu} = \,\frac{\pi}{2\sqrt{2}}\frac{I_{g}}{\sin( \frac{\varphi AB}{2} )},$

$I_{R,rms}\mspace{6mu} = \mspace{6mu}\frac{\pi}{2\sqrt{2n}}I_{load},$

$V_{Cr,rms}\mspace{6mu} = \mspace{6mu}\frac{2\sqrt{2n}}{\pi}V_{out}\sqrt{1 + \frac{1}{Q^{2}}},$

$I_{Cr,rms}\mspace{6mu} = \mspace{6mu}\frac{2\sqrt{2n}}{\pi}\frac{V_{out}}{Z_{o}}\sqrt{1 + \frac{1}{Q^{2}}}.$

From equations (25) - (28), it can be observed that for a givenoperating condition (I_(g), V_(out), φ_(AB)), the rms current of thesource side resonant inductor (L_(r)) is constant and independent ofload whereas, rms current in load side resonant inductor (L_(g)) isdirectly proportional to load (I_(load)). Root-mean-square (“RMS”)voltage and current of the resonant capacitor is also dependent on theload (Q).

From these rms current(s), referring to the circuit in FIG. 5(a), thevolt-ampere (“VA”) for the resonant tank components can be determined.The VA for L_(r) can be evaluated as:

VA_(Lr) = I_(t, rms)²Z_(o) = QP_(out^(,))

and the VA for L_(g) is evaluated to be:

$VA_{Lg} = I_{R,rms}^{2}Z_{o} = \frac{P_{out}}{Q}.$

Details of equations (29) and (30) are provided in the Appendix. The VAof the resonant capacitor can be found using equation (28) and isexpressed as:

VA_(Cr) = I_(Cr, rms)²Z_(o⋅)

Now, from the phasor diagram of FIG. 5(b) it can be seen that it andi_(R) in quadrature and ic_(r) is the phasor subtraction of it andi_(R). Hence, equation (28) can also be expressed as:

$I_{Cr,rms} = \sqrt{I_{t,rms}^{2} + I_{R,rms}^{2},}$

and using equation (32), equation (31) can be written as:

$VA_{Cr} = ( {I_{t,rms}^{2} + I_{R,rms}^{2}} )Z_{o} = ( {Q + \frac{1}{Q}} )P_{out}.$

It can be seen from equations (29), (30) and (33) that the VA of thetank capacitor is the sum of VA of the tank inductors. The total VA ofthe tank is calculated by summing up equations (29), (30) and (33) andis given by:

$VA_{tank} = 2( {Q + \frac{1}{Q}} )P_{out}.$

D. Design of Resonant Tank

To find the VA rating of the resonant tank, equation (34) can beevaluated at maximum output power (P_(out)__(max)). The normalized VArating (VA_(tank)__(norm)), with respect to P_(out_max) can be givenfrom equation (34) as:

$VA_{tank\_ norm} = 2( {Q_{Pout\_ max} + \frac{1}{Q_{Pout\_ max}}} ),$

where, Q_(pout_max) is the quality factor at maximum output power. Itcan be seen that the minimum value of VA_(tank_norm) from equation (35)is attained at:

$Q_{Pout\_ max} = \frac{1}{Q_{Pout\_ max}} = 1,$

and the minimum value of total tank VA rating is found out usingequations (34) and (36) and it is given as:

VA_(tank_min) = 4P_(out_max).

In order to design the tank elements, we need to start at equation (17)where Vout and I_(g) are known, but, n, Z₀ and φ_(AB) are to be decidedon. From the expression of Q in equation (7) and P_(out) from equation(10), the characteristic impedance of the tank can be written as:

$Z_{o} = \frac{8n^{2}}{\pi^{2}}\frac{V_{out}^{2}}{P_{out}Q\prime}$

and substituting Z₀ from equation (38) into equation (17), thetransformer turns ratio can be expressed as:

$n = \frac{P_{out}Q\sin( \frac{\varphi_{AB}}{2} )}{I_{g}V_{out}} \cdot$

In order to achieve minimum VA rating for the tank, the optimum value oftransformer turns ratio (n_(min-VA)) can be found by substituting Q =QPout_max = 1 from equation (36) into equation (39), and is given by:

$n_{\min\_ VA} = \frac{P_{out\_ max}\sin( \frac{\varphi_{AB}}{2} )}{I_{g}V_{out}}.$

The value of φAB is selected to be 120 degrees (“°”) which producesleast harmonic content at the output of the inverter with no triplenharmonics. This also provides sufficient margin from the maximumpossible control angle of 180° to support transient response. Afterdetermining the transformer turns ratio from equation (40), Z₀ isevaluated from equation (38) as:

$Z_{o} = \frac{ 8n_{\min}^{2} \__{VA}}{\pi^{2}}\frac{V_{out}^{2}}{P_{out\_ max}},$

and from equation (7), the tank element values can be calculated as:

$L_{r} = \frac{Z_{o}}{2\pi f_{o}} = \frac{Z_{o}}{2\pi f_{s}},$

$C_{r} = \frac{1}{2\pi f_{o}Z_{o}} = \frac{1}{2\pi f_{s}Z_{o}},$

$L_{g} = L_{r} = \frac{Zo}{2\pi f_{s}}.$

The ratings of the resonant tank elements are given in equations (25) -(28). With these design equations, the resonant tank and transformerturns ratio can be uniquely designed with minimum VA rating. However,designing the tank with minimum VA rating can result in discontinuouscurrent in the secondary diodes, which can possibly limit the range ofload for which converter output voltage can be regulated. This can beovercome by replacing diode bridge with an active bridge on thesecondary side of the converter, which is presented in the next sectionswith simulation and experimental results.

III. Dual Active Bridge LCL-T Resonant Converter

With the design method from Section II-D, the converter is designed fora system with 1 A input and 150 V output with a load range of 50 W to500 W. The designed parameters are listed in TABLE I. The plot ofquality factor and normalized tank VA rating for various transformerturns ratio is presented in FIG. 6 , where it can be seen that the tankVA is minimum at n = n_(min_VA) as per equation (40). In FIG. 6 , thequantities Q_(S), Q_(L) and Q_(tot) are defined as follows:

$Q_{s} = Q,Q_{L} = \frac{1}{Q},Q_{tot} = Q_{s} + Q_{L}.$

Details of equation (45) can be found in the Appendix.

TABLE I Resonant Tank Parameters L_(r) ( (µH) C_(r) (pF) L_(g) (µH) n194.4 208.5 194.4 2.9

A. Limitation With Diode-Bridge Rectifier

With the tank parameters listed in TABLE I, the converter of FIG. 2(a)is simulated in MATLAB®/ Piecewise Linear Electrical Circuit Simulation(“PLECS”®) and the steady state DC output voltage over the load range isplotted in FIG. 7 .

It can be seen from the top plot in FIG. 7 that the output voltage withsecondary side diode bridge rectifier does not stay constant,independent of load and increases in value as the load reduces. Sincethe tank is designed for minimum VA, the diodes operate in discontinuousconduction mode (“DCM”) due to low quality factor. This increase in Voutat light load will results in the control angle (φ_(AB)) going towardsits limit of 180°, to keep the output voltage at its desired value. Thiscan potentially hinder the load range of operation for which theconverter can regulate its output. In order to keep the diodes incontinuous conduction mode (“CCM”) over a load range, the tankcomponents have to be designed with higher VA rating which will increasethe size of the converter. Alternately, the converter can be designedwith lower nominal value of φ_(AB), considering the load range andmargin for component tolerances, but this will lead to higher componentstress (see equations 25 - 28) and losses.

B. Secondary Side Active Rectification

To operate the converter with wide range load regulation, it isdesirable to keep the secondary bridge in CCM. This is achieved byemploying an active bridge on the secondary side, as shown in FIG. 8(a).The modulation scheme for both primary and secondary bridges is depictedin FIG. 8(b), where φ_(DE) is the control angle between leg D and leg Eof secondary bridge, and φ_(AD) is the angle between leg A and leg D.

With a secondary active bridge, the current in L_(g) and in thetransformer secondary will be in CCM. To operate the secondary bridge atunity power factor (with FHA) and emulate the behavior of dioderectifier, the secondary bridge modulation angle φ_(DE) should be equalto 180°. From the phasor diagram shown in FIG. 5(b), the relationshipamong the three modulation angles [φ_(AS), φ_(AD), (φ_(DE) can be givenas:

$\varphi_{AD} = \frac{\varphi_{AB}}{2} + \frac{\pi}{2} - \frac{\varphi_{DE}}{2},$

and with φ_(DE) = 180° (e.g. Π radians), equation (46) becomes:

$\varphi_{AD} = \frac{\varphi_{AB}}{2}.$

The dual active bridge (“DAB”) LCL-T converter is also simulated inMATLAB/PLECS and the steady state DC output voltage at various load isplotted in FIG. 9 , with the tank parameters presented in TABLE I. Thisis also compared with the result achieved with a diode-bridge on thesecondary H-bridge. From the dotted plot in FIG. 9 , it can be seen thatthe steady state DC output voltage remains constant, independent ofload, as derived in equation (17), matching the analytical referenceplot.

In some embodiments, φ_(AB), φ_(AD), and φ_(DE) are set to fixed valuesdue to the load independence of the DAB LCL-T converter. In someembodiments, a control loop is used to maintain the output voltage Voutat a reference value. The control loop may be used to control any ofφ_(AB), φ_(AD), or φ_(DE). In some embodiments, φ_(DE) is set to 80°.The control loop compares the output voltage V_(out) with a referencesignal, which is fed into a compensator. The output of the compensatoris used to either control φ_(AB) or φ_(AD) (when φ_(DE) is set to 180°),which is then used to control modulation of the switches Q₁ - Q₈. Whereφ_(AB) is controlled, then equation (47) is used to set φ_(AD.) Whereφ_(AD) is controlled, then φ_(AB) is twice φ_(AD).

IV. Experimental Verification

A prototype hardware of the LCL-T converter of FIGS. 2(a) and 8(a) hasbeen built to verify the analysis presented so far with the tankparameters mentioned in TABLE I and additional details presented inTABLE II. The converter is designed to operate at 250 kHz switchingfrequency, which is the operating frequency of all the series connectedconverters used in this underwater DC current distribution network. Theprototype is designed for operation up to 500 W with 150 V DC outputvoltage and it is tested over 10:1 load range for experimentalverification. With operation at F = 1 and g = 1, both the primary andsecondary inverter(s) (e.g. H-bridges) operate with their fundamentalvoltage and current in phase which means that all the switches Q₁ - Q₈in the converter will not have ZVS by the tank current. On the primaryside inverter, leg B MOSFETs go through ZVS by tank current and anactive ZVS assisting circuit is used for leg A. Since the secondary sidebridge operates at unity power factor with fixed DC output voltage, afixed, passive inductor (L_(zvs)__(sec)) is used for ZVS of secondaryside MOSFETs. DC blocking capacitors (C_(DC_pri) and C_(DC_sec)) areused in both primary and secondary side H-bridges to block any DCcomponent of voltage arriving out of the H-bridges due to any componentmismatch. The value(s) of the DC blocking capacitor(s) is chosensignificantly higher than the resonant capacitor such that it does notimpact the overall capacitor seen from the resonant inductor terminalsand thus does not influence the resonant frequency of the tank.

TABLE II Converter Details Component/Parameter Value I_(g) (A) 1 V_(out)(V) 150 ƒ_(s) (kHz) 250 P_(out) (W) 50-500 Primary MOSFETs C2M1000170DC_(DCpri) µF) 0.23 L_(zvs pri) (µH) 50 Secondary Diode(s) FFSH2065B-F085Secondary MOSFETs IXFQ72N20X3 C_(DC) _(sec) (µF) 6.4 L_(zvs) _(sec) (µH)60

First, the converter was tested with a diode bridge rectifier and theresults are shown in FIG. 10 for 50 W and 500 W operation with φ_(AB) =120°. It can be seen from the ν_(DE) and i_(D) plot in FIG. 10(a) thatthe diode operates in DCM mode. Then the converter is tested with activerectifiers (e.g. switches in the secondary H-bridge creating a DAB) andthe steady state operating waveforms are shown in FIG. 11 for 50 W and500 W operation with φ_(AB) = 120°, keeping φ_(DE) = 180° and φ_(AD) =60°. In addition, the gate-source and drain-source voltage across thebottom switch(es) of all the H-bridge legs of DAB LCL-T are presented inFIG. 12 at minimum (50 W) and maximum (500 W) load. From this results inFIG. 12 , it can be seen that the drain-source voltage of the switch(es)falls to zero, before its gate-source voltage rises, confirming ZVSoperation over the entire load range. And, due to half wave symmetry inoperation, the top switches also go through ZVS turn on.

The steady state DC output voltage results with both a diode bridge andan active bridge secondary are plotted in FIG. 13 , versus the loadpower at fixed φ_(AB) = 120°. In FIG. 13 , the analytical plot is theanalytical reference (150 V) and the other traces represent the resultswith a diode rectifier and an active rectifier, respectively. It can beseen that the output voltage is not load independent for a diode bridge,whereas, with an active secondary bridge, the output voltage isrelatively constant over a 10:1 load range. The variation in V_(out)with an active secondary is between 149.1 V and 144.8 V, which is onlyabout 2.9%. The variation is due to non-idealities such as ESR ofcomponents. The results presented here are from 50 W to 500 W that istypical range of load seen by the converter. The converter will ideallymaintain its output voltage at the same level while the load power fallsfurther below 50 W. In practice, there could be deviation in outputvoltage from its ideal value, due to component tolerance, temperaturevariation etc., which will be taken care of by the controller for outputvoltage regulation. From the hardware results, the variation of V_(out)with a diode bridge is lower than predicted by simulation, which isattributed to the parasitic capacitance of the diodes. Once the diodeturns off, the diode capacitance resonates with L_(g), which can be seenfrom ν_(DE) and i_(D) plot in FIG. 10(a).

The variation of control angle (φ_(AB)) needed to keep the outputvoltage regulated at 150 V is plotted in FIG. 14 , where again theanalytical line shows the analytical prediction and the other tracesrepresent the results with a diode rectifier and an active rectifier. Itcan be seen from these plots that variation in φ_(AB) is quite large foroperation with a diode bridge, whereas the variation is only 5.4° withactive bridge on the secondary, over the 10:1 load range.

The DAB LCL-T resonant converter was also tested with load transient atits output with a fixed control angle (φ_(AB) = 115°) and the result isshown in FIG. 15 . The output load is varied from 350 W to 400 W andthen back to 350 W. In FIG. 15 , the input DC voltage (V_(in)) iscaptured by CH2, output current (I_(load)) is captured by CH1 and theoutput voltage (V_(out)) is captured by CH4. It can be observed fromthis result that even under load transient the output voltage returns tothe same value at steady state.

The rms values of tank inductor current and capacitor voltages aremeasured from the oscilloscope captures at different loads and arecompared with the analytical values derived in equations (25) - (27).The comparison is shown in FIG. 16 , where analytical values are plottedin solid lines and the measured values are shown in correspondingcircles of the same component. The top plot in FIG. 16 compares the rmscurrent in the tank inductors and the bottom plot compares the resonantcapacitor voltage. The result depicts a good match between experimentalresult and analysis.

The analytically evaluated power loss in different components of the DABLCL-T converter is presented in FIG. 17 , at full load operatingcondition. These losses are used for components and heat sink designwith natural cooling. The efficiency of the converter, operating atfixed control angle of φ_(AB) = 120°, with active and passive bridges onthe secondary side, are shown in FIG. 18 . It can be seen that theconverter operates with a higher efficiency with an active secondarybridge due to the lower conduction loss. The peak efficiency isapproximately 96%.

V. Converter Insights

In underwater DC distribution systems, a constant current source is usedto power to multiple, series-connected converters to achieve robustnessagainst voltage drop over the cable length and cable faults. However,powering from a current source brings in various challenges in converterdesign. Addressing these challenges, as discussed above, it is shownthat an LCL-T resonant DC-DC converter can be designed to achieve aload-independent, constant DC output voltage characteristic when poweredfrom a constant DC current source. Detailed modeling, analysis anddesign are presented for this converter. With analysis, simulation andhardware results, it is shown that diode bridge rectification on theoutput side of the converter imposes a challenge on low Q (VA rating)design and the use of active bridge overcomes this limitation. Amodulation scheme for the DAB LCL-T resonant converter is presented foroverall operation of the converter with minimum VA rating for the tankcomponents, the isolation transformer, and the H-bridges. Finally, ahardware prototype is developed and tested for a system with 1 A inputcurrent, 150 V output voltage, operating at a switching frequency of 250kHz, over a load range of 50 W to 500 W. Results obtained from hardwareexperiments confirm the analysis with a good match between analyticalexpressions and experimentally obtained values.

VII. Bidirectional Converter

As stated above, underwater power distribution network used in oceanobservatory system uses constant current distributed through a longdistance trunk cable. In some embodiments, multiple power branchingunits (“PBUs”) are connected in series to tap power from the DC currentfeed to deliver required voltage or current to their respective loads,as shown in FIG. 19(a). These PBUs can deliver power to a fixedstationary load regulating it output current or output voltage or can becharging an underwater autonomous vehicle wirelessly.

Some of these PBUs deliver power to critical loads where redundant,identical DC-DC power converter modules are used within a PBU, as shownin FIG. 19(b). The DC-DC converters within a PBU are connected to theinput/output of the PBU through relay networks to select appropriateconverter module(s) for seamless power delivery to the critical load. Inaddition, there is an auxiliary source housed within the PBU(s) foruninterrupted power for the critical loads. As shown in FIG. 20(a),under normal conditions, power flows from the constant current source tothe load through one of the DC-DC converter modules to provide constantvoltage to the load, while the other module is bypassed and kept idle.In case of fault in the line, in some embodiments, the relay networksreconfigure the PBU with a connection shown in FIG. 20(b) where thesecond DC-DC module converts the power from auxiliary voltage source toa constant current drive feeding the first converter which then deliverspower to the load, regulating its output voltage. Hence the powerconverter modules need to be capable of converting a current source to avoltage source in forward direction of power flow and a voltage sourceto a current drive in reverse direction of power flow. While the exampleabove provides a rationale to develop a bidirectional DAB LCL-Tconverter, this bidirectional DAB LCL-T converter may be used for manyother situations where one side is constant current and the other isconstant voltage.

In embodiments described below, a detailed analysis is presented for anisolated DAB LCL-T resonant converter with generalized three anglemodulation for the active bridges, having current source input inforward power flow and voltage source input in reverse power flow. Themodulation angles are specifically controlled for converters used inconstant current distribution systems and the resonant tank andtransformer turn ratio are designed for minimization of VA ratings ofthe converter components.

VIII. Steady State Modeling and Analysis

The DAB LCL-T converter topology is detailed in FIG. 21(a). In forwarddirection of power flow, power is transferred from DC current sourceI_(g) to load R_(L2), to regulate converter’s output DC voltage V₂, asdepicted in FIG. 21(a). Whereas, in reverse direction of power flow,power from DC voltage source V_(g) flows to R_(L1), regulating DCcurrent I₁ as its output in FIG. 21(a). In FIG. 21(a), MOSFETs Q₁ - Q₄form primary side H-bridge that translates DC voltage V₁ into an ACquasi-square wave ν_(AB) through symmetrical phase shift modulation,with leg A leading leg B by an angle φ_(AB), as shown in FIG. 21(b).Similarly, MOSFETs Q₅ - Q₈ form secondary side H-bridge between DCvoltage V₂ and AC quasi-square wave ν′_(DE), modulated by angle φ_(DE),with leg D leading leg E whose time domain waveform is also shown inFIG. 21(b). As shown in FIG. 21(b), the two H-bridges are separated byangle φ_(AD), which is the phase shift between positive rising edge ofν_(AB) and ν′_(DE). As a matter of nomenclature definition, withreference to the modulation waveform presented in FIG. 21(b), any angle(φ_(XY)) used in the analysis is defined as:

φ_(XY) = φ_(Y) − φ_(X).

The resonant tank is formed by capacitor C_(r) and two equal valuedinductors L_(r) and L_(g), transferring power between the two H-bridgesthrough a n:1 isolation transformer. Capacitors C₁ and C₂ filter outhigh frequency signals at the DC side of the H-bridges. With fundamentalharmonics approximation (“FHA”), the converter shown in FIG. 21(a) canbe redrawn as the equivalent circuit shown in FIG. 22(a), for the steadystate analysis, and it is assumed that all the components are ideal andlossless.

A. AC Equivalent Circuit Analysis

The fundamental AC equivalent circuit of the loaded LCL-T resonant tank,reflected to transformer primary side, is shown in FIG. 22(b) where vsand is represent the source side fundamental AC voltage and current,respectively and ν_(L) and i_(L) represent load side fundamental ACvoltage and current, respectively. In the circuit shown in FIG. 22(b),the impedance Z_(e) is the equivalent complex impedance seen at the ACside of the H-bridge on the load side and is represented as:

Z_(e) = |Z_(e)|∠φ_(e),

where φ_(e) is the angle between ν_(L) and i_(L). The details ofderiving this impedance are provided in Appendix A.

Since the converter’s switching frequency (ƒ_(s)) is same as itsresonant frequency (ƒ_(o)), the circuit in FIG. 22(b) is symmetric andcan be analyzed irrespective of power flow direction from I_(g) toR_(L2) or from V_(g) to R_(L1). The circuit quantities used in theanalysis are defined as

$\begin{matrix}{f_{o} = \frac{1}{2\pi\sqrt{L_{r}C_{r}}},} & {Z_{o} = \sqrt{\frac{L_{r}}{C_{r}},}} & {L_{g} = L_{r},} & {F = \frac{f_{s}}{f_{o}} = 1.}\end{matrix}$

From the equivalent circuit of FIG. 22(b), the input impedance of theloaded tank, seen from the source side can be derived as:

$Z_{s} = jZ_{o} + \frac{- jZ_{o}( {Z_{e} + jZ_{o}} )}{- jZ_{o} + Z_{e} + jZ_{o}} = \frac{Z_{o}^{2}}{| Z_{e} |}\angle - \varphi_{e}.$

Using Z_(S) from equation (51), the AC source current is can be foundas:

$i_{s} = \frac{V_{S}}{Z_{S}} = V_{S}\frac{| Z_{e} |}{Z_{o}^{2}}\angle\varphi_{e},$

where Vs is the amplitude of vs. The load side AC current i_(L) can bederived as:

$i_{L} = \frac{V_{s}}{Z_{o}}\angle - \frac{\pi}{2}.$

The voltage across and current through resonant capacitor C_(r) can bederived as:

$v_{Cr} = V_{L}cos( \varphi_{e} )\sqrt{1 + \frac{1}{Q_{Z}^{2}}}\angle - tan^{- 1}Q_{z},$

$i_{Cr} = \frac{V_{L}}{Z_{o}}cos( \varphi_{e} )\sqrt{1 + \frac{1}{Q_{Z}^{2}}}\angle\mspace{6mu} tan^{- 1}( \frac{1}{Q_{Z}} ),$

where V_(L) is the amplitude of _(VL) and Q_(z) is defined as:

$Q_{Z} = \frac{| Z_{e} |cos( \varphi_{e} )}{Z_{o} + | Z_{e} |sin( \varphi_{e} )}.$

From the derivations in equations (52) - (55), the phasor diagram of theAC equivalent circuit in FIG. 22(b) can be drawn as shown in FIG. 23 ,with vs taken as reference and φ_(SL) defined as the phase angle betweenνs and ν_(L) and is related to φ_(e) by:

$\varphi_{e} = \frac{\pi}{2} - \varphi_{SL}.$

The source and load power in the circuit of FIG. 22(b) are expressed as:

$P_{S} = \frac{V_{S}^{2}}{2}\frac{| Z_{e} |}{Z_{o}^{2}}cos( \varphi_{e} ),$

$P_{L} = \frac{V_{L}^{2}}{2| Z_{e} |}cos( \varphi_{e} ).$

The power transfer from source to load, through the resonant tank, interms of source and load side AC voltages can be given as:

$P_{T} = \frac{V_{S}V_{L}}{2Z_{o}}cos( \varphi_{e} ).$

From equation (60), the maximum power transfer for a given resonant tankwill occur with maximum values of V_(s) and V_(L) and at φ_(e) = 0 andthis value can be given in terms of DC voltage(s) V₁ and V₂ as:

$| P_{max} | = \frac{8n}{\pi^{2}}\frac{V_{1}V_{2}}{Z_{o}},$

and the set of modulation angle(s) at which maximum power transferoccurs is given as:

$\begin{matrix}{\varphi_{AB} = \mspace{6mu}\pi,} & {\varphi_{DE} = \mspace{6mu}\pi,} & | {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} |\end{matrix} = \frac{\pi}{2}.$

From the analysis and phasor diagram of the AC equivalent circuitpresented in this section, the following key properties of LCL-Tresonant tank can be observed

-   1. For power transfer from the source to the load, source voltage    (v_(S)) will lead the load voltage (v_(L)).-   2. The load current (i_(L)) always lags source voltage (v_(S)) by    90°, for any load impedance.-   3. An inductive impedance on the load side (i_(L) lagging _(VL))    will reflect as capacitive on the source side (i_(S) leading v_(S))    and vice versa.

B. DC Input - Output Relationship

From the analysis of the AC resonant circuit established in previoussection, the relationship of input and output DC quantities can be nowderived from the equivalent circuit modeled in FIG. 22(a). Thefundamental component of AC voltage from the two H-bridges are given as:

$V_{AB,1} = \frac{4}{\pi}V_{1}sin( \frac{\varphi_{AB}}{2} ),$

$V_{DE,1} = \frac{4n}{\pi}V_{2}\mspace{6mu} sin( \frac{\varphi_{DE}}{2} ).$

For forward power flow, substituting equations (63) and (64) intoequations (58) and (59) and equating the source and load side power thefollowing relationship is established:

$\frac{V_{1}\mspace{6mu} sin( \frac{\varphi_{AB}}{2} )}{nV_{2}\mspace{6mu} sin( \frac{\varphi_{DE}}{2} )} = \frac{Z_{o}}{| Z_{e} |}.$

The magnitude of AC load impedance is given as:

$\begin{matrix}{| Z_{e} | = \frac{8n^{2}}{\pi^{2}}R_{L2}cos} & ( \varphi_{e} ) & {sin^{2}} & {( \frac{\varphi_{DE}}{2} ),}\end{matrix}$

where R_(L2) is the load resistance on the DC output side. From thecircuit in FIG. 22(a), the DC power input (P_(in)) and power output(Pout) can be given as:

$\begin{matrix}{P_{in} = V_{1}I_{1},} & {P_{out} = \frac{V_{2}^{2}}{R_{L2}}.}\end{matrix}$

With lossless power conversion, equating the input and output DC powerfrom equation (67), the input DC voltage can be expressed as:

$V_{1} = \frac{V_{2}^{2}}{I_{1}R_{L2}}.$

Substituting the V₁ from equation (68) and |Z_(e)| from equation (66)into equation (65), the expression of V₂ for forward power flow can bederived as:

$V_{2} = \frac{\pi^{2}}{8n}\frac{I_{1}Z_{o}}{sin( \frac{\varphi_{AB}}{2} )sin( \frac{\varphi_{DE}}{2} )cos( \varphi_{e} )}.$

The value of φ_(e) can be found from equation (57) with φ_(SL) evaluatedfor forward power, from the modulation waveform in FIG. 21(b) as:

$\varphi_{SL\_ F} = \varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}.$

Finally, substituting equations (70) and (57) into equation (69), the DCoutput voltage V₂ can be evaluated as:

$V_{2} = \frac{\pi^{2}}{8n}\frac{I_{1}Z_{o}}{sin( \frac{\varphi_{AB}}{2} )sin( \frac{\varphi_{DE}}{2} )sin( {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} )}.$

From equation (71) it can be observed that with I₁ = I_(g) i.e. withconstant DC current source, beneficially the output DC voltage of theconverter becomes independent of load. The input voltage, however, willbe dependent on load and is found out by plugging in V₂ from equation(71) in equation (68) and is given by:

$V_{1} = \frac{I_{1}}{R_{L2}}\lbrack {\frac{\pi^{2}}{8n}\frac{Z_{o}}{sin( \frac{\varphi_{AB}}{2} )sin( \frac{\varphi_{DE}}{2} )sin( {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} )}} \rbrack^{2}.$

The DC input current to the converter (I₁) and AC input to the resonanttank (it) are related through the primary side H-bridge by:

$I_{1} = \langle i_{1} \rangle = \frac{2I_{t}}{\pi}sin( \frac{\varphi_{AB}}{2} )cos( \varphi_{e} ),$

where It is the amplitude of it and average value of current i₁ isrepresented by <i₁>. For forward power flow, I₁ = I_(g) and thus thetank input current can be written from equation (73) and using equation(52) as:

$i_{t} = \frac{\pi}{2}\frac{I_{g}}{sin( \frac{\varphi_{AB}}{2} )cos( \varphi_{e} )}\angle\varphi_{e}.$

Similarly, the load side tank AC current i_(R) can be expressed, usingthe phase information from equation (53), as:

$i_{R} = \frac{\pi}{2n}\frac{V_{2}}{R_{L2}sin( \frac{\varphi_{DE}}{2} )cos( \varphi_{e} )}\angle - \frac{\pi}{2}.$

Following similar approach, the equations of signals for reverse powerflow can be derived which are tabulated in Table III along with theequations for forward power flow. The phasor diagram for the tank ACsignals are also presented in the Table III, for both forward andreverse power flow, taking ν_(AB,1) and ν_(DE,1) taken as reference,respectively. The power flow direction from primary to secondary andvice versa is dependent of the phase angle (φ_(PS)) between ν_(AB,1) andν_(DE,1) which can be found out from the modulating waveform shown inFIG. 21(b) and is given by:

$\varphi_{PS} = \varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}.$

For forward power flow where I₁ and I₂ are positive, φ_(ps) is withinthe range [0,_(Π)] and in reverse power flow where I₁ and I₂ are bothnegative, (φ_(PS) is within the range [—_(Π), 0]. Thus, the relationshipbetween (φ_(SL) and φ_(PS) is given by:

$\varphi_{SL} = | \varphi_{PS} | = | {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} |,$

and from equation (57) the relationship between φ_(e) and modulationangles is given by:

$\varphi_{e} = \frac{\pi}{2} - | \varphi_{PS} | = \frac{\pi}{2} - | {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} |.$

The variation of φ_(SL) and φ_(e) are plotted against φ_(PS) in FIG. 24along with the variation in normalized power from primary to secondaryusing equation (61), for φ_(AB) = φ_(DE) = Π, which is defined as:

P_(PS_norm) = sin(φ_(PS)).

The relationships established between DC input and output quantities andthe derived AC quantities of the resonant tank, in either direction ofpower flow, are used for choosing the converter operating point anddesign of components in the following section.

TABLE III Converter Signals VALUE FORWARD POWER REVERSE POWER SourceI_(g) V_(g) Load (R_(L)) R_(L2) R_(L1) V₁$\frac{I_{g}}{R_{L2}}\lbrack {\frac{\pi^{2}}{8n}\frac{Z_{o}}{\sin( \frac{\varphi_{AB}}{2} )\sin( \frac{\varphi_{DE}}{2} )\sin( {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} )}} \rbrack^{2}$|I₁|R_(L1) I₁ I_(g)$\frac{8}{\pi^{2}}\frac{nV_{g}}{Z_{o}}\sin( \frac{\varphi_{AB}}{2} )\sin( \frac{\varphi_{DE}}{2} )\sin( {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} )$V₂$\frac{\pi^{2}}{8n}\frac{I_{g}Z_{o}}{\sin( \frac{\varphi_{AB}}{2} )\sin( \frac{\varphi_{DE}}{2} )\sin( {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} )}$V_(g) I₂ $\frac{V_{2}}{R_{L2}}$ $\begin{array}{l}{- R_{L1}V_{g}\lbrack \frac{8n}{\pi^{2}Z_{o}} )\sin( ( \frac{\varphi_{AB}}{2} ) )\sin( \frac{\varphi_{DE}}{2} )\sin( {\varphi_{AD} - \frac{\varphi_{AB}}{2}} )} \\{+ ( ( \frac{\varphi_{DE}}{2} ) \rbrack^{2}}\end{array}$ φ_(e)$\frac{\pi}{2} - ( {\varphi_{AD} - \frac{\varphi_{AB}}{2} + ( \frac{\varphi_{DE}}{2} )} )$$\frac{\pi}{2} + ( {\varphi_{AD} - \frac{\varphi_{AB}}{2}} ) + ( \frac{\varphi_{DE}}{2} )$Z_(e)$\frac{8n^{2}}{\pi^{2}}R_{L2}\cos(\varphi_{e})\sin^{2}( \frac{\varphi_{DE}}{2} )\angle\varphi_{e}$$\frac{8}{\pi^{2}}R_{L1}\cos(\varphi_{e})\sin^{2}( \frac{\varphi_{AB}}{2} )\angle\varphi_{e}$v_(S)$v_{AB,1} = \frac{4}{\pi}V_{1}\sin( \frac{\varphi_{AB}}{2} )\angle 0$$v_{DE,1} = \frac{4n}{\pi}V_{g}\sin( {\frac{\varphi_{DE}}{2}\angle 0} )$v_(L) $\begin{array}{r}{V_{DE,1} = \frac{4n}{\pi}V_{2}\sin( \frac{\varphi_{DE}}{2} )\angle} \\{- ( {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} )}\end{array}$$V_{AB,1} = \frac{4}{\pi}V_{1}\sin( \frac{\varphi_{AB}}{2} )\angle( {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} )$i_(S)$i_{t} = \frac{\pi l_{g}}{2\sin( \frac{\varphi_{AB}}{2} )\cos( \varphi_{e} )}\angle\varphi_{e}$$i_{R} = \frac{4}{\pi}| I_{1} |\frac{R_{L1}}{Z_{o}}\sin( \frac{\varphi_{AB}}{2} )\angle( {\pi + \varphi_{e}} )$i_(L)$i_{R} = \frac{\pi}{2n}\frac{V_{2}}{R_{L2}\sin( \frac{\varphi_{DE}}{2} )\cos( \varphi_{e} )}\angle - \frac{\pi}{2}$$i_{t} = \frac{4n}{\pi}\frac{V_{g}}{Z_{o}}\sin( \frac{\varphi_{DE}}{2} )\angle\frac{\pi}{2}$v_(Cr)$\frac{4n}{\pi}V_{2}\sin( \frac{\varphi_{DE}}{2} )\cos( \varphi_{e} )\sqrt{1 + \frac{1}{Q_{z}^{2}}}\angle - \tan^{- 1}Q_{z}$$\frac{4}{\pi}| I_{1} |R_{L1}\sin( \frac{\varphi_{AB}}{2} )\cos(\varphi_{e})\sqrt{1 + \frac{1}{Q_{z}^{2}}}\angle - \tan^{- 1}Q_{z}$i_(Cr)$\frac{4n}{\pi}\frac{V_{2}}{Z_{o}}\sin( \frac{\varphi_{DE}}{2} )\cos(\varphi_{e})\sqrt{1 + \frac{1}{Q_{z}^{2}}}\angle\tan^{- 2}( \frac{1}{Q_{z}} )$$\frac{4}{\pi}| l_{1} |\frac{R_{L1}}{Z_{o}}\sin( \frac{\varphi_{AB}}{2} )\cos( \varphi_{e} )\sqrt{1 + \frac{1}{Q_{z}^{2}}}\angle\tan^{- 1}( \frac{1}{Q_{z}} )$Phasor diagram

IX. Design of the Converter

From the analysis presented in the previous section, converter gain forforward (G_(F)) and reverse (G_(R)) power are expressed as:

$G_{F} = \frac{V_{2}}{I_{g}} = \frac{\frac{\pi^{2}}{8n}Z_{o}}{sin( \frac{\varphi_{AB}}{2} )sin( \frac{\varphi_{DE}}{2} )sin( {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} )},$

$G_{R} = \frac{I_{1}}{V_{g}} = \frac{sin( \frac{\varphi_{AB}}{2} )sin( \frac{\varphi_{DE}}{2} )sin( {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} )}{\frac{\pi^{2}}{8n}Z_{o}}.$

It can be observed from equations (80) and (81) that the magnitude(s) ofthe gain(s) are reciprocal to each other, i.e.:

$| G_{F} | = \frac{1}{| G_{R} |}.$

which means that for a given resonant tank (Z_(o)) and transformer (n)the input to output ratio can be achieved with same set of modulationangle [φ_(AB), φ_(DE), |φ_(AD) - φ_(AB)/2 + φ_(DE)/2|], with source ACvoltage leading the load side voltage. And thus, with a designedmodulation angle set, the tank and transformer can be designedirrespective of power flow direction.

A. Modulation Angle

In forward power flow, when the converter is fed from a DC currentsource, a non-zero φ_(e) makes the input impedance seen by the sourceH-bridge either inductive or capacitive. This brings in a restriction onminimum power operation of the converter for which the output can beregulated. So, to eliminate such limitation, φ_(e) is made to be zerowhich, from equation (78) gives:

$| {\varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}} | = \frac{\pi}{2}.$

Now, with the condition established in equation (83), control of outputcan be done through φ_(AB) or φ_(DE). In this application, since thesecondary side of this converter has higher current compared to theprimary side, φ_(DE) is set to its maximum value of 180° to keep thedevice current stress low for the secondary H-bridge and control of theconverter is done through φ_(AB).

Finally, the nominal operating value of φ_(AB) is chosen to be 120°which eliminates any triplen harmonic content out of the primaryH-bridge. This also keeps good margin from maximum possible value ofφ_(AB) as 180°, for transients. With known φ_(AB) and φ_(DE), φ_(AD) isfound out using equation (83) and the set of modulation angles forforward power is given as:

$\varphi_{AB} = 120^{{^\circ}},\mspace{6mu}\mspace{6mu}\varphi_{DE} = 180^{{^\circ}},\mspace{6mu}\mspace{6mu}\varphi_{AD} = \frac{\varphi_{AB}}{2},$

whereas, for reverse power, the set is given as:

$\varphi_{AB} = 120^{{^\circ}},\mspace{6mu}\mspace{6mu}\varphi_{DE} = 180^{{^\circ}},\mspace{6mu}\mspace{6mu}\varphi_{AD} = \frac{\varphi_{AB}}{2} - 180^{{^\circ}}.$

B. Derivation of Converter Signals at Operating Point

With the selected operating set of modulating angles in equations (84)and (85), the resultant signals of the converter from Table III can besimplified to the signals tabulated in Table IV where the terms Z_(e)and Q_(z) are modified to R_(e) and Q, respectively. From equation (56),Q is expressed as:

$Q = \frac{R_{e}}{Z_{o}}.$

From the expressions of the tank AC signals and phasor diagram presentedin Table IV, it can be observed that at the selected operatingcondition, both the H-bridges now operate at unity power factor (UPF),in terms of their fundamental AC voltage and current. Also, from thesederivations, rms currents in the tank components can be calculated fromtheir signal amplitude which are used to calculate the VA of the tank todesign the resonant tank and transformer turn ratio for lowest VArating.

C. Design of Resonant Tank and Transformer

The VA of the resonant tank can be expressed in terms of rms currentthrough L_(r), L_(g) and C_(r) as:

V A_(Tank) = (I_(t, rms)² + I_(R, rms)² + I_(Cr, rms)²)Z_(o).

Since the current through the resonant capacitor is phasor subtractionof i_(t) and i_(R) which are in quadrature to each other, the rmscurrents through the tank elements can be related as:

$I_{Cr,rms} = \sqrt{I_{t,rms}^{2} + I_{R,rms}^{2}} \cdot$

Substituting equation (88) in equation (87) the tank VA is evaluated as:

VA_(Tank) = 2(I_(t, rms)² + I_(R, rms)²)Z_(o).

Using the equations of i_(t), i_(R) and V₂ from Table IV and Q fromequation (86), the tank VA expression from equation (89) can be furthersimplified to:

$VA_{Tank} = 2( {Q + \frac{1}{Q}} )P_{load},$

TABLE IV Converter Signals at the Operating Condition VALUE FORWARDPOWER REVERSE POWER V₁$\frac{I_{g}}{R_{L2}}\lbrack {\frac{\pi^{2}}{8n}\frac{Z_{o}}{\sin( \frac{\varphi_{AB}}{2} )}} \rbrack^{2}$|I₁|R_(L1) I₁ I_(g)$- \frac{8n}{\pi^{2}}\frac{V_{g}}{Z_{o}}\sin( \frac{\varphi_{AB}}{2} )$V₂$\frac{\pi^{2}}{8n}\frac{I_{g}Z_{o}}{\sin( \frac{\varphi_{AR}}{2} )}$V_(g) I₂ $\frac{V_{2}}{R_{L2}}$$- R_{L1}V_{g}\lbrack {\frac{8n}{\pi^{2}Z_{o}}\sin( \frac{\varphi_{AB}}{2} )} \rbrack^{2}$R_(e) $\frac{n^{2}8}{\pi^{2}}R_{L2}$$\frac{8}{\pi^{2}}R_{L1}\sin^{2}( \frac{\varphi_{AB}}{2} )$v_(S)$v_{AB,1} = \frac{4}{\pi}V_{1}\sin( \frac{\varphi_{AB}}{2} )\angle 0$$v_{DE,1} = \frac{4n}{\pi}V_{g}\angle 0$ v_(L)$v_{DE,1} = \frac{4n}{\pi}V_{2}\angle - \frac{\pi}{2}$$V_{AB,1} = \frac{4n}{\pi}V_{1}\sin( \frac{\varphi_{AB}}{2} )\angle - \frac{\pi}{2}$i_(S)$i_{t} = \frac{\pi}{2}\frac{l_{g}}{\sin( \frac{\varphi_{AB}}{2} )}\angle 0$$i_{R} = \frac{\pi}{2}\frac{| l_{1} |}{\sin( \frac{\varphi_{AB}}{2} )}Q\angle\pi$i_(L) $i_{R} = \frac{4n}{\pi}\frac{V_{2}}{QZ_{o}}\angle - \frac{\pi}{2}$$i_{t} = \frac{4n}{\pi}\frac{V_{g}}{Z_{o}}\angle\frac{\pi}{2}$ v_(Cr)$\frac{4n}{\pi}V_{2}\sqrt{1 + \frac{1}{Q^{2}}}\angle - \tan^{- 1}Q$$\frac{\pi}{2}\frac{Z_{o}| l_{1} |}{\sin( \frac{\varphi_{AB}}{2} )}\sqrt{1 + Q^{2}}\angle - \tan^{- 1}Q$i_(Cr)$\frac{4n}{\pi}\frac{V_{2}}{Z_{o}}\sqrt{1 + \frac{1}{Q_{2}}}\angle\tan^{- 1}1( \frac{1}{Q} )$$\frac{\pi}{2}\frac{| l_{1} |}{\sin( \frac{\varphi_{AB}}{2} )}\sqrt{1 + Q^{2}}\angle\tan^{- 1}( \frac{1}{Q} )$Phasor diagram

where P_(load) represents the output load. The expression from isnormalized with respect to P_(load) and expressed as:

$VA_{Tank\_ norm} = 2( {Q + \frac{1}{Q}} ).$

From the expression of V₂ in Table IV, Z_(o) can be expressed as:

$Z_{o} = \frac{8n}{\pi^{2}}\frac{V_{2}sin( \frac{\varphi_{AB}}{2} )}{1g},$

and substituting this value in equation (86), Q can be expressed as:

$Q = \frac{nI_{g}V_{2}}{P_{load}sin( \frac{\varphi_{AB}}{2} )}.$

The tank VA rating is found under the maximum load condition (P_(load) =P_(load_max)) and from equation (91) it can be seen that the normalizedVA of the tank would be minimum when Q = 1, under maximum load. PluggingQ = 1 in equation (93), the optimum value of transformer turn ratio(n_(opt)) can be expressed as:

$n_{opt} = \frac{P_{load\_ max}sin( \frac{\varphi_{AB}}{2} )}{V_{2}I_{g}}.$

Using equation (94) in equation (92), Z_(o) can be evaluated from whichthe values of resonant tank components can be calculated as:

$L_{r} = \frac{Z_{o}}{2\pi f_{o}} = \frac{Z_{o}}{2\pi f_{s}},$

$C_{r} = \frac{1}{2\pi f_{o}Z_{o}} = \frac{1}{2\pi f_{s}Z_{o}},$

$L_{g} = L_{r} = \frac{Z_{o}}{2\pi f_{s}}.$

The normalized VA rating of the tank is plotted against transformer turnration (n) in FIG. 25 where it can be seen that the minimum value oftank VA occurs at n = n_(opt).

In some embodiments, φ_(AB), φ_(AD), and φ_(DE) are set to fixed valuesdue to the load independence of the DAB LCL-T converter. In otherembodiments, a voltage control loop is used to maintain the outputvoltage v_(out) at a reference value for forward power and a currentcontrol loop is used to control current at the input-side of the primaryH-bridge for reverse power. The voltage control loop may be used tocontrol any of φ_(AB), φ_(AD), or φ_(DE). In some embodiments, φ_(DE) isset to 180°. The voltage control loop compares the output voltagev_(out) with a reference signal, which is fed into a compensator. Theoutput of the compensator is used to either control φ_(AB) or φ_(AD)(when φ_(DE) is set to 180°), which is then used to control modulationof the switches Q₁ - Q₈. Where φ_(AB) is controlled, then

$\varphi_{AD} = \frac{\varphi_{AB}}{2}$

φ_(AD) is used to set φ_(AD) for forward power. Where φ_(AD) iscontrolled, for forward power then φ_(AB) is twice φ_(AD).

For reverse power, the current control loop may be used to control anyof φ_(AB), φ_(AD), or φ_(DE). In some embodiments, φ_(DE) is set to180°. The current control loop compares the input current I_(g) with areference signal, which is fed into a compensator. The output of thecompensator is used to either control φ_(AB) or φ_(AD) (when φ_(DE) isset to 180°), which is then used to control modulation of the switchesQ₁ - Q₈. Where φ_(AB) is controlled, then

$\varphi_{AD} = \frac{\varphi_{AB}}{2} - 180^{\circ}$

is used to set φ_(AD) for reverse power. Where φ_(AD) is controlled,then φ_(AB) = 2(φ_(AD) + 180°) is used to set φ_(AB) for reverse power.

D. ZVS Assistance

FIG. 31 is a schematic block diagram of the circuit of FIG. 21(a) withZVS circuits. With the H-bridges of the converter designed to operate atUPF (FHA), all the MOSFETs will not have zero voltage switching (ZVS)and hence ZVS assisting circuits are needed. The secondary H-bridgeoperates at UPF with φ_(DE) = 180°, resulting in zero current turn onand turn off, in either direction of power flow. So, a fixed inductor(L_(zvs_sec)) based passive ZVS assisting circuit is used across leg Dand leg E, since the bridge operates with fixed DC voltage V₂. On theprimary side H-bridge, which operates with φ_(AB) = 120°, tank current(it) does ZVS for MOSFETs in lagging leg, for forward power and leadingleg, for reverse power. For the other leg in the primary bridge, anactive ZVS assisting circuit with an inductor (L_(zvs_pri)) and ahalf-bridge is used to achieve ZVS over the wide load range, since theDC voltage (V₁) for primary bridge varies with load.

X. Experimental Verification

A prototype hardware has been built to verify the analysis presented inthe last sections whose details are presented in TABLE V. DC blockingcapacitors (C_(DC_pri) and C_(DC_sec)) are used in both primary andsecondary side H-bridges to block any DC component of voltage arrivingout of the inverters due to any component non-idealities.

For forward power transfer, the converter is tested with 1 A constant DCcurrent source with modulation angle set from equation (84) for a loadrange of 50 W to 500 W and the steady state waveforms of the H-bridgevoltage and current are shown in FIG. 26(a) and FIG. 26(b) for a load of50 W and 500 W, respectively. In FIG. 26(c) and FIG. 26(d), same set ofwaveforms are shown for reverse power flow from a 150 V constant DCvoltage source, with modulation angle set presented in equation (85). InFIG. 26 , CH1 shows the waveform of v_(AB), CH2 is for it, CH3 is forv′_(DE) and current i′_(R) is shown in CH4. It can be seen from thesewaveforms that for forward power transfer, fundamental components ofv_(AB) and it are in phase and v′_(DE) and i′_(R) are in phase withv_(AB) leading v′_(DE) by 90°, as per the derivations and phasor diagrampresented in Table IV. Whereas, in reverse power transfer, thecorresponding pair of voltage and current are 180° out of phase withv′_(DE) leading v_(AB) by 90°, conforming to the derivations and phasordiagram presented in Table IV.

TABLE V Bidirectional Converter Details Component/Parameter Value I_(g)/ |I₁| (A) 1 V₂ / V_(out) (V) 150 ƒ_(s) (kHz) 250 L_(r) (µH) 194.4 C_(r)(pF) 208.5 L_(g) (µH) 194.4 n 2.9 P_(load) (W) 50-500 Primary MOSFETsC2M1000170D C_(DC pri) (µF) 0.23 L_(zvs pri) (µH) 50 Secondary MOSFETsIXFQ72N20X3 C_(DC sec) (µF) 6.4 L_(zvs) _(sec) (µH) 60

The steady state DC output voltage (V₂), with fixed control angle ofφ_(AB) = 120°, is plotted over the load range in FIG. 27(a), for forwardpower transfer. And the DC output current (|I₁|) with fixed controlangle of φ_(AB) = 120° is plotted in FIG. 27(b), for reverse powertransfer operation. In FIG. 27 , the analytical lines represent theanalytical reference and the other lines show the experimental data. Itcan be seen from these plots that the DC output is almost constant over10:1 load range, making them fairly independent of load. The smallvariation in the DC output is within around 3% for forward power andwithin 4% for reverse power transfer operation which is due tonon-idealities such as ESR of components, effects of dead time etc.,which were ignored in analysis.

The variation of control angle (φ_(AB)) needed to keep the outputregulated at the desired value is also checked for this converteroperating in both directions of power flow. The experimental data isplotted in FIG. 28 where the forward power transfer and reverse powertransfer are depicted along with the analytical reference. Theexperimental data range of φ_(AB) is within the control boundary of theconverter and the variation is only within 8°, over the entire loadrange, for either direction of power flow.

The rms current in the tank inductors and rms voltage across theresonant capacitor are also measured in hardware, from the oscilloscopecaptures, for the entire load range in both direction of power flow andare compared to their analytical values. The comparisons are shown inFIG. 29(a) for forward power and in FIG. 29(b) for reverse poweroperations. In FIG. 29 , the analytical results from Table IV areplotted in solid line whereas the measured data are shown incorresponding dots of same variable. The top plot in FIG. 29(a) and FIG.29(b) compares the rms current (it and i_(R)) in the tank inductors andthe bottom plot compares the rms value of resonant capacitor voltage(v_(Cr)). The plots in FIG. 29 depicts a good match between analysis andresults obtained from the hardware experiments.

The converter efficiency while regulating its DC output at its desiredvalue of 150 V for forward power transfer and 1 A for reverse powertransfer, are shown in FIG. 30 with a peak efficiency around 96%.

XI. Conclusion

In underwater DC distribution system, constant current source basedsystem provides benefit over constant voltage source due to robustnessagainst voltage drop along cable length and cable fault. In such system,power converter capable of bi-directional power flow are required forcritical loads. With detailed analysis presented herein, it is shownthat a bidirectional DAB LCL-T resonant converter which can provide loadindependent DC output voltage from constant DC current source in forwardpower flow and load independent DC current at its output when fed from aconstant DC voltage source in reverse power flow, is well suited forsuch critical underwater PBUs. Starting with steady state modeling andanalysis, incorporating generalized three angle modulation, it ispresented how the converter can be designed with proper modulation angleset in order to operate the converter with minimum VA rating for theH-bridges, resonant tank and transformer. A hardware prototype has beenbuilt and tested for a load range of 50 W to 500 W, converting 1 Asource to 150 V output in forward direction and 150 V DC voltage sourceto 1 A drive to load in reverse power, operating at a switchingfrequency of 250 kHz. And the test results show a good match betweenanalysis and experimental data in terms of the steady state DC output,rms values of tank signals and phasor relationship among the AC signals.

XII. Appendix A. Derivation of Tank Input Impedance

With reference to the AC equivalent circuit shown in FIG. 3(b), theimpedances of individual tank components are given by:

$\begin{array}{l}{X_{Lr} = 2\pi f_{s}L_{r} = FZ_{o},X_{Lg} = gFZ_{o},} \\{X_{Cr} = \frac{1}{2\pi f_{s}C_{r}} = \frac{Z_{o}}{F},}\end{array}$

where, Z_(o) and F are as defined in equation (7). Now, the tank inputimpedance can be derived as:

$Z_{in} = jFZ_{o} + \frac{- j\frac{Z_{o}}{F}( {R_{e} + jgFZ_{o}} )}{- j\frac{Z_{o}}{F} + ( {R_{e} + jgFZ_{o}} )^{\prime}}$

which can be simplified to:

$Z_{in} = jFZ_{o} + \frac{Z_{o}( {Q + jgF} )}{( {1 - gF^{2}} ) + jFQ^{\prime}}$

where, Q is defined in equation (7). The expression of Z_(in) from (A.3)can further be expanded to be expressed in the form:

$Z_{in} = \frac{Z_{o}}{\lbrack {( {1 - gF^{2}} )^{2} + F^{2}Q^{2}} \rbrack}\lbrack {Z_{R} + jZ_{I}} \rbrack,$

where, Z_(R) and Z_(I) are expressed as:

$\begin{array}{l}{Z_{R} = Q( {1 - gF^{2}} )( {1 - F^{2}} )^{2} + ( {1 + g} )QF^{2} - gQF^{4},} \\{Z_{I} = ( {1 + g} )F( {1 - gF^{2}} ) - gF^{3}( {1 - gF^{2}} ) - FQ^{2}( {1 - F^{2}} )^{2}.}\end{array}$

When operating at resonance, i.e. at F = 1, Z_(in) from (A.4) can besimplified to:

$Z_{in{|{F = 1})}} = \frac{Z_{o}}{\sqrt{Q^{2} + ( {1 - g} )^{2}}}\angle tan^{- 1}( \frac{1 - g}{Q} ).$

and the power factor, cos(φ_(in)), can be given as:

$\cos( \varphi_{in} )| {}_{F = 1} ) = \frac{Q}{\sqrt{Q^{2} + ( {1 - g} )^{2}}}.$

Substituting (A.6) and (A.7) into equation (12), the DC output voltagecan be expressed as:

$ V_{out} \middle| {}_{F = 1} = \frac{\pi^{2}}{8n}\frac{Z_{o}I_{g}}{\sin( \frac{\varphi_{AB}}{2} )}. $

Further, when g = 1, i.e. with L_(g) = L_(r), Z_(in) and cos(φ_(in)),are given as:

$ Z_{in} \middle| {}_{F = 1,g = 1} = \frac{Z_{o}^{2}}{R_{e}}\angle 0^{o},\mspace{6mu}\mspace{6mu}\cos( \varphi_{in} ) \middle| {}_{F = 1,g = 1} = 1 $

B. Derivation of Tank Quantities

In this section, the tank signals are derived from the equivalentcircuit shown in FIG. 3(a) and FIG. 5(a). The input current (i_(t)) tothe resonant tank from the inverter is shown in FIG. B1 and itsrectified DC side current (i_(in)) is shown on the right hand side ofthe same figure. Since the average value of i_(in) comes from the DCsource (I_(g)), the amplitude of it can be found out through:

$I_{\text{g}} = \langle i_{in} \rangle = \frac{2}{\pi}I_{t}\sin( \frac{\varphi AB}{2} ),$

where, <x> denotes average of x over its period. Since the converter isoperating at F = 1 and g = 1, v_(AB,1) and it are in phase and thus,using (B.1), the current in the resonant inductor L_(r) can be expressedas:

$i_{t}(t) = I_{t}\cos( {\omega_{s}t} ) = \frac{\pi}{2}\frac{I_{g}}{\sin( \frac{\varphi_{AB}}{2} )}\cos( {\omega_{s}t} ).$

The amplitude of load side resonant inductor current (i_(R)) isevaluated from the circuit in FIG. 5(a) with amplitude of v_(o,1) fromequation (8) and is given as:

$I_{R} = \frac{| v_{o,1} |}{R_{e}} = \frac{\frac{4n}{\pi}V_{out}}{\frac{8n^{2}}{\pi^{2}}R_{load}} = \frac{\pi}{2n}I_{load}.$

Now, from the circuit of FIG. 5(a), i_(R) can be expressed in terms ofit as:

$i_{R} = i_{t}\frac{- jZ_{o}}{- jZ_{o} + R_{e} + jZ_{o}} = - j\frac{Z_{o}}{R_{e}}i_{t},$

which means that i_(R) lags it by 90° and thus, using (B.3), i_(R)(t)can be express as:

$i_{R}(t) = I_{R}\cos( {\omega_{2}t - \frac{\pi}{2}} ) = \frac{\pi}{2n}I_{load}\,\cos\,( {\omega_{2}t - \frac{\pi}{2}} ).$

The voltage across the resonant capacitor can be found from FIG. 5(a) aswell and is derived as:

$\begin{array}{l}{\nu_{C}{}_{r} = i_{R}(R_{e} + jZ_{o}) = \lbrack {\frac{4n}{\pi}\frac{V_{out}}{R_{e}}\angle( {- \frac{\pi}{2}} )} \rbrack R_{e}( {1 + \frac{j}{Q}} )} \\{= \frac{4n}{\pi}V_{out}\sqrt{1 + \frac{1}{\text{Q}^{2}}\angle}\lbrack {- \frac{\pi}{2} + \tan^{- 1}( \frac{1}{\text{Q}} )} \rbrack.}\end{array}$

Using trigonometric identity, it can be shown that:

$\tan^{- 1}Q = \,\frac{\pi}{2} - \tan^{- 1}( \frac{1}{Q} )$

Hence, from (B.6) the resonant capacitor voltage is given as:

$v_{Cr}(t) = \frac{4n}{\pi}V_{out}\sqrt{1 + \frac{1}{Q^{2}}}\cos( {\omega_{s}t - \tan^{- 1}Q} ).$

The current through C_(r) is evaluated as:

$i_{Cr} = \frac{v_{Cr}}{- jZ_{o}} = \frac{4n}{\pi}\frac{V_{out}}{Z_{o}}\sqrt{1 + \frac{1}{Q^{2}}}\angle( {- \tan^{- 1}Q\,\text{+}\frac{\pi}{2}} ).$

With the identity shown in (B.7), current in the resonant capacitor isexpressed as:

$i_{Cr}(t) = \frac{4n}{\pi}\frac{V_{out}}{Z_{o}}\sqrt{1 + \frac{1}{Q^{2}}}\cos( {\omega_{s}t + \tan^{- 1}\,\frac{1}{Q}} ).$

The VA of L_(r) is evaluated as:

$VA_{Lr} = I_{t,rms}^{2}Z_{o} = ( \frac{\pi}{2\sqrt{2}} )^{2}\frac{I_{g}^{2}}{( {\sin( \frac{\varphi AB}{2} )} )^{2}}Z_{o}.$

Using the expression of V_(out) from equation (17), (B.11) can bewritten as:

$VA_{Lr} = \frac{8n^{2}}{\pi^{2}Z_{o}}R_{load}\frac{V_{out}^{2}}{R_{load}} = \frac{R_{e}}{Z_{o}}P_{out} = QP_{out} \cdot$

VA of L_(g) is evaluated as:

$\begin{array}{l}{VA_{LG} = I_{R,rms}^{2}Z_{o} = ( {\frac{\pi}{2\sqrt{2n}}I_{load}} )^{2}Z_{o} = \frac{\pi^{2}}{8n^{2}}( \frac{V_{out}}{R_{load}} )^{2}Z_{o}} \\{= \frac{Z_{o}}{\frac{8n^{2}}{\pi^{2}}R_{load}}( \frac{V_{out}^{2}}{R_{load}} ) = \frac{1}{\frac{R_{e}}{Z_{o}}P_{out}}P_{out} = \frac{P_{out}}{\text{Q}} \cdot}\end{array}$

The quality factors presented in equation (45) are derived here fromtheir basic definition. The quality factor of the load side resonantinductor L_(g) is derived as:

$Q_{L} = 2\pi\frac{E_{Lg\_ pkfo}}{P_{out}},$

where, E_(Lg_pk) is the peak energy stored in L_(g) and is given as:

$E_{Lg\_ pk} = \frac{1}{2}L_{g}( {\sqrt{2}I_{R,rms}} )^{2} = L_{g}I_{R,rms}^{2},$

and from the equivalent circuit of FIG. 5(a), P_(out) can be given as:

P_(out) = I_(R, rms)²R_(e)⋅

Substituting (B.15) and (B.16) into (B.14) and using definition ofω_(o), Z_(o) and Q from equation (7), Q_(L) can be expressed as:

$Q_{L} = 2\pi f_{o}\frac{L_{g}}{R_{e}} = \frac{Z_{o}}{R_{e}} = \frac{1}{Q}.$

Similarly, the quality factor of the source side resonant inductor L_(r)can be derived as:

$Q_{s} = 2\pi\frac{E_{Lr\_ pkfo}}{P_{out}},$

where, E_(Lr_pk) is the peak energy stored in L_(r) and is given as:

$E_{Lr\_ pk} = \frac{1}{2}L_{r}( {\sqrt{2}I_{t,rms}} )^{2} = L_{r}I_{t,rms}^{2}.$

Using the relationship between i_(t) and i_(R) from (B.4) and using(B.19) andequation (7), (B.18) can be further expressed as:

$Q_{s} = 2\pi f_{o}\frac{L_{r}}{R_{e}}( \frac{R_{e}}{Z_{o}} )^{2} = \frac{R_{e}}{Z_{o}} = Q\text{.}$

It can be observed from (B.17) and (B.20) that Q_(S) and Q_(L) areinverse of each other which means if source side inductor current ismore sinusoidal (less in harmonic content), the load side inductorcurrent will be more non-sinusoidal (more harmonic content). This can beobserved from the results shown in FIG. 10(a), where at light load, thewaveform of i_(t) is more sinusoidal than waveform of i_(R) (transformerreflected i_(D)) when Q_(S) = 10 and Q_(L) = 0.1.

C. Tolerance Analysis

The variation of output voltage due to tolerances in tank componentvalues are shown in FIG. C1 using the analytical expression of V_(out)from equation (12) and Z_(in) from equation (13). In FIG. C1 , theanalytical percentage variation in V_(out) is plotted in sloid lines forvariation in L_(r), C_(r) and L_(g), with one of them varied at a time,keeping the remaining two fixed at their nominal value(s). The toleranceresults obtained from MATLAB-PLECS simulation is also shown in the sameplot, using dots of corresponding color. It can be seen from this plotin FIG. C1 that there is a little variation in output voltage forvariation in tank inductors. However, variation in resonant capacitorhas a dominant effect on variation in V_(out). The results plotted inFIG. C1 are for the lowest load of P_(out) = 50 W, where the value of Qis maximum and has highest influence on V_(out) due to componenttolerance. With use of a class I ceramic capacitor (C0G, NP0), which isstable over temperature and voltage bias, the capacitance tolerance iswithin ±5% which translates to variation in V_(out) within ±6%, from theresult plotted in FIG. C1 . This can be taken care of by margin inmodulation angle of φ_(AB), (120° to 180°). Further, since the toleranceis prominent at light loads, active shunt current control circuit canalso be utilized at the input source with slight drop of light loadefficiency.

D. Derivation of H-bridge Impedance

An H-bridge controlled through phase shift modulation angle φ_(I) isshown in FIG. D1(a) whose voltage and current signals on the DC side areV_(DC), I_(DC) and ν_(AC), i_(AC) on the AC side. The voltage andcurrent waveforms of this H-bridge are shown in FIG. D1(b) with ν_(AC,1)being the fundamental component of ν_(AC). With ν_(AC,1) as reference,the AC side quantities of the H-bridge are expressed as:

$v_{AC,1}(t) = \frac{4}{\pi}V_{DC}sin( \frac{\varphi_{I}}{2} )\,\, sin( {\omega_{s}t} ),$

i_(AC)(t)=  I_(AC)  sin(ω_(s)t − φ_(AC)),

where φ_(AC) is the phase shift angle between ν_(AC,1) and i_(AC). FromFIG. D1(b), the DC side current I_(DC) can be evaluated from the AC sidecurrent as:

$I_{DC} = \frac{1}{\frac{\pi}{\omega_{s}}}{\int_{\frac{\pi}{2} - \frac{\varphi I}{2}}^{\frac{\pi}{2} - \frac{\varphi I}{2}}{i_{AC}(t)dt = \frac{2}{\pi}I_{AC}sin( \frac{\varphi_{I}}{2} )cos( \varphi_{AC} ).}}$

The impedance seen from the AC side of the H-bridge is expressed as:

Z_(AC)=  |Z_(AC)|∠φ_(AC),

where |Z_(AC)| is calculated as:

$| Z_{AC} \middle| \,\, = \,\,\frac{| v_{AC,1} |}{| i_{AC} |} = \frac{8}{\pi^{2}}R_{DC}\,\, cos( \varphi_{AC} )\,\, sin^{2}( \frac{\varphi_{I}}{2} ),\, $

where R_(DC) is the load resistance on the DC side of the H-bridge. Theimpedance in (D.4) can also be expressed in cartesian form which isgiven as:

Z_(AC)  =  R_(AC) + jX_(AC),

where R_(AC) and X_(AC) are the real and imaginary part of Z_(AC),respectively and are defined as:

R_(AC)  =  |Z_(AC)|  cos(φ_(AC)),  X_(AC)  =  |Z_(AC)|  sin(φ_(AC)).

E. Sensitivity to Component Variation

The variation of DC output voltage, for forward power plow and DC outputcurrent, for reverse power flow are plotted in FIG. E1 , for variationin each tank components, one at a time, keeping the other two elementsat their nominal value. The results shown in FIG. E1 is from simulationcarried out in MATLAB-PLECS where the solid lines represent thepercentage variation in the DC output for forward power flow and thedotted lines are for reverse power flow with respect to percentagevariation in L_(r), C_(r) and L_(g). From this plot it can be seen thatthe DC output is not sensitive to the variation in inductor values butare strongly dependent on variation in resonant capacitor. However,since class I ceramic capacitors (C0G, NP0), stable over voltage biasand temperature, are used as the resonant capacitor, the tolerance incapacitance is within ±5% resulting in a variation in output within ±6%.This variation can be taken care of by the control range of φ_(AB).

The present invention may be embodied in other specific forms withoutdeparting from its spirit or essential characteristics. The describedembodiments are to be considered in all respects only as illustrativeand not restrictive. The scope of the invention is, therefore, indicatedby the appended claims rather than by the foregoing description. Allchanges which come within the meaning and range of equivalency of theclaims are to be embraced within their scope.

What is claimed is:
 1. A power converter comprising: a primary H-bridgecomprising semi-conductor switches; an LCL-T section comprising a firstinductor L_(r) with a first end connected to a first terminal A of theprimary H-bridge, a capacitor C_(r) connected between a second end ofthe first inductor L_(r) and a second terminal B of the primaryH-bridge, and a second inductor L_(g) with a first end connected to thesecond end of the first inductor L_(r); a transformer with a primaryside connected between a second end of the second inductor L_(g) and thesecond terminal B of the primary H-bridge; a secondary H-bridgecomprising semi-conductor switches with an input connected to asecondary side of the transformer; and an output capacitor C_(ƒ)connected across output terminals of the secondary H-bridge, wherein theprimary H-bridge is fed by a direct current (“DC”) constant currentsource and the output terminals of the secondary H-bridge are connectedto a load and an output voltage of the secondary H-bridge regulated tomaintain a constant DC output voltage.
 2. The power converter of claim1, wherein a switching frequency of the switches of the primary H-bridgeand the secondary H-bridge is selected to be within 15 percent of aresonant frequency of the LCL-T section.
 3. The power converter of claim2, wherein a ratio g of the first inductor L_(r) and the second inductorL_(g) is set to be within a range of 0.2 to 5 (g = 0.2 to 5).
 4. Thepower converter of claim 1, wherein the switches of the primary H-bridgeare arranged in a leg A and a leg B, the switches of the secondaryH-bridge are arranged in a leg D and a leg E and wherein: the switchesof the primary H-bridge are operated with symmetrical phase shiftmodulation with leg A leading leg B by an angle φ_(AB); the switches ofthe secondary H-bridge are operated with symmetrical phase shiftmodulation with leg D leading leg E by an angle φ_(DE); an angle betweenleg A and leg D is angle φ_(AD); and the output voltage of the secondaryH-bridge is maintained at a constant voltage by controlling angleφ_(AB), angle φ_(DE), and angle φ_(AD).
 5. The power converter of claim4, wherein a relationship between angle φ_(AB), angle φ_(DE), and angleφ_(AD) is:$\varphi_{AD} = \frac{\varphi_{AB}}{2} + \frac{\pi}{2} - \frac{\varphi_{DE}}{2}$.
 6. The power converter of claim 5, wherein angle φ_(DE) is 180 degreesand a relationship between angle φ_(AB) and angle φ_(AD) is:$\varphi_{AD} = \frac{\varphi_{AB}}{2}$ .
 7. The power converter ofclaim 6, wherein one of: angle φ_(AB) is controlled as a function of theoutput voltage of the secondary H-bridge and angle φ_(AD) is controlledto be half the angle φ_(AB); and angle φ_(AD) is controlled as afunction of the output voltage of the secondary H-bridge and angleφ_(AB) is controlled to be twice the angle φ_(AD).
 8. The powerconverter of claim 1, wherein power flow is bidirectional.
 9. The powerconverter of claim 8, wherein the switches of the primary H-bridge arearranged in a leg A and a leg B, the switches of the secondary H-bridgeare arranged in a leg D and a leg E and wherein: the switches of theprimary H-bridge are operated with symmetrical phase shift modulationwith leg A leading leg B by an angle φ_(AB); the switches of thesecondary H-bridge are operated with symmetrical phase shift modulationwith leg D leading leg E by an angle φ_(DE); an angle between leg A andleg D is angle φ_(AD); and a power flow direction from the primaryH-bridge to the secondary H-bridge is dependent on a phase angle φ_(PS),which is:$\varphi_{PS} = \varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}$.
 10. The power converter of claim 9, wherein φ_(PS) is within the range[0, π] for forward power flow where input current I₁to the primaryH-bridge and output current I₂ from the secondary H-bridge are positive,and φ_(PS) is within the range [-π, 0] for reverse power flow where I₁and I₂ are both negative.
 11. The power converter of claim 9, whereinangle φ_(DE) is 180 degrees and a relationship between angle φ_(AB) andangle φ_(AD) is: $\varphi_{AD} = \frac{\varphi_{AB}}{2}$ for forwardpower flow; and $\varphi_{AD} = \frac{\varphi_{AB}}{2} - 180{^\circ}$for reverse power flow.
 12. The power converter of claim 11, wherein oneof: for forward power flow, angle φ_(AB) is one of set to a fixed valueor controlled as a function of the output voltage of the secondaryH-bridge and angle φ_(AD) is controlled to be half the angle φ_(AB); forreverse power flow, angle φ_(AB) is one of set to a fixed value orcontrolled as a function of the input current to the primary H-bridgeand angle φ_(AD) is controlled to be$\varphi_{AD} = \frac{\varphi_{AB}}{2} - 180{^\circ};$ for forward powerflow, angle φ_(AD) is one of set to a fixed value or controlled as afunction of the output voltage of the secondary H-bridge and angleφ_(AB) is controlled to be twice the angle φ_(AD); and for reverse powerflow, angle φ_(AD) is one of set to a fixed value or controlled as afunction of the input current to the primary H-bridge and angle φ_(AB)is controlled to be φ_(AB) = 2(φ_(AD) + 180°).
 13. The power converterof claim 1, wherein a turns ratio n of the transformer is set at anoptimal turns ratio n_(opt):$n_{opt} = \frac{P_{load\_ max}sin( \frac{\varphi_{AB}}{2} )}{V_{2}I_{g}},$where the switches of the primary H-bridge are operated with symmetricalphase shift modulation with leg A leading leg B by an angle φ_(AB);P_(load_max) is a maximum load condition; I_(g) is a DC constant sourcecurrent; and V₂ is a constant output voltage of the secondary H-bridge.14. The power converter of claim 1, further comprising an inputcapacitor C_(in) connected across input terminals of the primaryH-bridge.
 15. A power converter comprising: a primary H-bridgecomprising four semi-conductor switches, wherein two of the switches arein leg A with terminal A between the switches in leg A and two of theswitches are in leg B with terminal B between the switches in leg B,terminal A and terminal B forming an output of the primary H-bridge; anLCL-T section comprising a first inductor L_(r) with a first endconnected to terminal A, a capacitor C_(r) connected between a secondend of the first inductor L_(r) and terminal B, and a second inductorL_(g) with a first end connected to the second end of the first inductorL_(r); a transformer with a primary side connected between a second endof the second inductor L_(g) and terminal B, the transformer comprisinga turns ratio n; a secondary H-bridge comprising semi-conductor switcheswith an input connected to a secondary side of the transformer, whereintwo of the switches are in leg D with terminal D between the twoswitches of leg D and two of the switches are in leg E with terminal Ebetween the two switches of leg E, terminal D and terminal E forming anoutput of the secondary H-bridge; and an output capacitor C_(ƒ)connected across terminal D and terminal E, wherein the primary H-bridgeis fed by a direct current (“DC”) constant current source and terminalsD and E are connected to a load and an output voltage across terminals Dand E is regulated to maintain a constant DC output voltage.
 16. Thepower converter of claim 15, wherein a switching frequency of theswitches of the primary H-bridge and the secondary H-bridge is selectedto be within 15 percent of a resonant frequency of the LCL-T section,and wherein a ratio g of the first inductor L_(r) and the secondinductor L_(g) is within a range of 0.2 to 5 (g = 0.2 to 5).
 17. Thepower converter of claim 15, wherein the switches of the primaryH-bridge are arranged in a leg A and a leg B, and the switches of thesecondary H-bridge are arranged in a leg D and a leg E and wherein: theswitches of the primary H-bridge are operated with symmetrical phaseshift modulation with leg A leading leg B by an angle φ_(AB); theswitches of the secondary H-bridge are operated with symmetrical phaseshift modulation with leg D leading leg E by an angle φ_(DE); an anglebetween leg A and leg D is angle φ_(AD); and the output voltage of thesecondary H-bridge is maintained at a constant voltage by controllingangle φ_(AB), angle φ_(DE), and angle φ_(AD), wherein a relationshipbetween angle φ_(AB), angle φ_(DE), and angle φ_(AD) is:$\varphi_{AD} = \frac{\varphi_{AB}}{2} + \frac{\pi}{2} - \frac{\varphi_{DE}}{2}$.
 18. The power converter of claim 17, wherein angle φ_(DE) is 180degrees and a relationship between angle φ_(AB) and angle φ_(AD) is$\varphi_{AD} = \frac{\varphi_{AB}}{2},$ and wherein one of: angleφ_(AB) is controlled as a function of the output voltage of thesecondary H-bridge and angle φ_(AD) is controlled to be half the angleφ_(AB); and angle φ_(AD) is controlled as a function of the outputvoltage of the secondary H-bridge and angle φ_(AB) is controlled to betwice the angle φ_(AD).
 19. The power converter of claim 15, whereinpower flow is bidirectional, and wherein the switches of the primaryH-bridge are arranged in a leg A and a leg B, the switches of thesecondary H-bridge are arranged in a leg D and a leg E and wherein: theswitches of the primary H-bridge are operated with symmetrical phaseshift modulation with leg A leading leg B by an angle φ_(AB); theswitches of the secondary H-bridge are operated with symmetrical phaseshift modulation with leg D leading leg E by an angle φ_(DE); an anglebetween leg A and leg D is angle φ_(AD); and a power flow direction fromthe primary H-bridge to the secondary H-bridge is dependent on a phaseangle φ_(PS), which is:$\varphi_{PS} = \varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}$.
 20. A bidirectional power converter comprising: a primary H-bridgecomprising four semi-conductor switches, wherein two of the switches arein leg A with terminal A between the switches in leg A and two of theswitches are in leg B with terminal B between the switches in leg B,terminal A and terminal B forming an output of the primary H-bridge; anLCL-T section comprising a first inductor L_(r) with a first endconnected to terminal A, a capacitor C_(r) connected between a secondend of the first inductor L_(r) and terminal B, and a second inductorL_(g) with a first end connected to the second end of the first inductorL_(r); a transformer with a primary side connected between a second endof the second inductor L_(g) and terminal B, the transformer comprisinga turns ratio n; a secondary H-bridge comprising semi-conductor switcheswith an input connected to a secondary side of the transformer, whereintwo of the switches are in leg D with terminal D between the twoswitches of leg D and two of the switches are in leg E with terminal Ebetween the two switches of leg E, terminal D and terminal E forming anoutput of the secondary H-bridge; and an output capacitor C_(ƒ)connected across terminal D and terminal E, wherein the primary H-bridgeis fed by a direct current (“DC”) constant current source and terminalsD and E are connected to a load and an output voltage across terminals Dand E is regulated to maintain a constant DC output voltage, wherein theswitches of the primary H-bridge are arranged in a leg A and a leg B,and the switches of the secondary H-bridge are arranged in a leg D and aleg E and wherein: the switches of the primary H-bridge are operatedwith symmetrical phase shift modulation with leg A leading leg B by anangle φ_(AB); the switches of the secondary H-bridge are operated withsymmetrical phase shift modulation with leg D leading leg E by an angleφ_(DE); an angle between leg A and leg D is angle φ_(AD); and a powerflow direction from the primary H-bridge to the secondary H-bridge isdependent on a phase angle φ_(PS), which is:$\varphi_{PS} = \varphi_{AD} - \frac{\varphi_{AB}}{2} + \frac{\varphi_{DE}}{2}$.